15^ 


GIFT  OF 


O. 


''  ■'-Lt^^tc  Lovy  Jo      O'^t'JeX; 


"""^> 


^'^>- 

^^^J^ 


/ 


TWENTIETH   CENTURY  TEXT-BOOKS 


TWENTIETH   CENTURY   TEXT-BOOKS 


A  HIGH  SCHOOL  ALGEBRA 


BY 


J.   W.   A.   YOUNG,   Ph.D, 

I  i 

ASSOCIATE    PROFESSOR   OF    THE    PEDAGOGY    OF    MATHEMATICS 
THE    UNIVERSITY    OF    CHICAGO 

AND 

LAMBERT   L.  JACKSON,    Ph.D. 

FORMERLY    PROFESSOR    OF    MATHEMATICS,    STATE    NORMAL 
SCHOOL,    BROCKPORT,    NEW  YORK 


D.    APPLETON    AND    COMPANY 

NEW    YORK  CHICAGO 


COPTRiaHT,   1913,   BT 

D.  APPLETON  AND  COMPANY. 


PREFACE 

This  volume  presents  a  full  high-school  course  in  elementary 
algebra  and  contains  all  the  topics  given  in  the  standard  year- 
and-a-half  courses.  It  is  adapted  to  the  prevailing  practice 
of  teaching  elementary  algebra  in  two  courses  —  a  full-year 
course  followed  by  a  half-year  course.  The  first  twenty-three 
chapters  contain  all  the  work  required  in  any  standard  one- 
year  course,  and  the  remaining  ten  chapters  comprise  a  subse- 
quent half-year  course,  reviewing  and  extending  the  elementary 
processes,  fractions,  factoring,  exponents,  and  methods  of  solv- 
ing equations,  before  any  new  topics  are  given.  The  result  is 
a  single  volume  adapted  to  a  continuous  one-and-a-half-year 
course,  or  to  a  course  in  which  geometry  intervenes  between 
first-year  and  second-year  algebra.  It  is  especially  suited  to 
the  latter  plan,  because  Chapters  XXIV  and  XXV  furnish  the 
review  necessary  for  thoge  pupils  who  take  the  divided  course. 
Moreover,  the  treatment  of  quadratic  equations,  radicals,  expo- 
nents, ratio,  proportion,  variation,  and  graphs  in  the  second  as 
well  as  in  the  first  year's  work,  gives  the  greatest  flexibility  to 
the  use  of  the  book.  For  example,  if,  for  the  purposes  of  a 
short  course,  one  or  more  of  the  later  chapters  were  omitted 
from  the  first  year's  work,  the  chapters  in  the  second  year's 
work  would  supply  material  on  the  subjects  omitted. 

In  whatever  manner  the  study  of  geometry  and  algebra  is 
alternated,  the  student  acquires  little  knowledge  of  the  met- 
rical properties  of  geometry  during  the  first  year.  For  this 
reason,  the  authors  have  used  in  their  problems  only  the  most 
obvious  of  these  properties,  and  have  given  in  a  carefully  pre- 
pared supplement  the  more  difficult  properties  to  which  algebra 
may  be  applied. 

264193 


vi  PREFACE 

Each  important  process  of  algebra  is  immediately  applied  to 
the  solving  of  equations.  This  plan  serves  not  only  to  secure 
the  pupiPs  interest,  but  reveals  to  him  the  utility  of  algebra. 

Great  pains  have  been  taken  to  supply  ample  practice  work, 
and  the  authors  have  given  under  the  more  important  topics, 
such  as  equations,  factoring,  highest  common  factor,  fractions 
and  exponents,  a  greater  number  of  exercises  than  will  be  re- 
quired by  any  one  class.  In  fact,  there  have  been  included  as 
many  exercises  and  problems  as  a  text-book  of  reasonable  size 
will  admit. 

Particular  attention  has  been  given  to  the  grading  of  the 
exercises  and  problems,  and,  for  convenience  in  checking  the 
results,  the  exercises  have  been  so  constructed  that  the  answers 
are  not  more  complex  than  the  purpose  of  the  exercises  actually 
requires.  The  authors  have  followed  the  criterion  that  every 
principle  should  be  exemplified  with  the  minimum  of  calculation. 

Among  the  features  that  contribute  to  the  teachableness  of 
the  book  are  the  Historical  Notes.  These  brief  sketches,  de- 
scribing  the  origin  of  some  of  the  more  important  topics  of 
algebra,  tend  to  stimulate  the  pupil's  interest,  and  the  accom- 
panying biographical  notes  and  portraits  of  famous  mathemati- 
cians serve  further  to  humanize  the  subject.  No  attempt  has 
been  made  to  give  a  connected  account  of  the  development  of 
algebra  even  in  outline;  these  notes  will  serve  their  purpose  if 
they  create  a  desire  to  read  some  standard  work  on  the  history 
of  mathematics. 

Other  aids  which  teachers  will  appreciate  are  the  inductive 
developments,  the  cross  references,  illustrative  problems,  meth- 
ods of  testing  results,  careful  statement  of  rules,  topical  and 
logical  arrangement,  definite  classification,  the  frequent  reviews, 
and  the  summaries  of  the  theoretic  chapters. 

The  authors  wish  to  express  their  gratitude  for  the  assistance 
rendered  by  those  whu  read  the  manuscript  and  the  proof 
sheets.  For  valuable  constructive  suggestions  in  preparing 
the  manuscript,  they  are  indebted  to  Mr,  Allen  H.  Knapp,  of 
the  Central  High  School,  Springfield,  Mass.,  and  Mr.  Julius 
J.  H.  Hayn,  of  the  Masten  Park  High  School,  Buffalo,  N.  Y. ; 


PREFACE  vii 

while  for  efficient  aid  in  reading  the  proofs,  they  owe  much 
to  Mr.  Matthew  R.  McCann,  of  the  English  High  School, 
Worcester,  Mass.,  and  Mr.  William  H.  Wentworth,  of  the 
Cass  Technical  High  School,  Detroit,  Mich.  For  the  portraits 
of  famous  mathematicians  reproduced  in  this  volume,  they  are 
indebted  to  the  generosity  of  Professor  David  Eugene  Smith, 
of  Teachers  College,  Columbia  University,  New  York  City, 
who  placed  at  their  disposal  his  unique  collection. 

THE  AUTHORS. 


CONTENTS 


OHAPTBB  PAOB 

I.  Literal  Notation  and  its  Uses  ....  1 

II.  Definitions  of  Elementary  Terms     ...  8 

III.  The  Equation 21 

IV.  Relative  Numbers 31 

V.     Addition 46 

VI.     Subtraction 54 

VII.    Equations 65 

VIII.     Multiplication 75 

IX.     Division 84 

X.    Equations 94 

XI.    Type  Products 102 

XII.    Factoring .        .109 

XIII.  Equations 128 

XIV.  Factors  and  Multiples 135 

XV.    Fractions 141 

XVI.     Equations 167 

XVII.  Ratio,  Proportion,  and  Variation      .        .        .  181 

XVIII.  Graphs  of  Linear  Equations       ....  198 

XIX.  Systems  of  Linear  Equations      ....  211 

XX.    Involution  and  Evolution 239 

XXI.  Radicals  and  Exponents        .        .        .        .        .  252 

XXII.     Quadratic  Equations 269 

XXIII.  Systems  of  Quadratic  Equations        .        .        .  282 

"XXrV^T  Keview  and  Extension  of  Processes         .        .  291 

XXV.     Equations 827 

XXVI.    Exponents  and  Roots 355 

XXVIL  Logarithms        ........  379 

iz 


CONTENTS 


CHAPTER  PAGB 

XXVIII.  Imaginary  and  Complex  Numbers       .        .        .  397 

XXIX.  Graphs  of  Quadratic  Equations         .        .        .  407 

XXX.     Quadratic  Equations 421 

XXXI.  Proportion,  Variation,  and  Limits     .        .         .  447 

XXXII.     Series .  465 

XXXIII.  Geometric  Problems  for  Algebraic  Solution  491 

INDEX '505 


A  HIGH   SCHOOL  ALGEBRA 

CHAPTER  I 
LITERAL  NOTATION  AND  ITS  USES 

1.  Numbers  represented  by  Letters.  In  arithmetic,  num- 
bers are  represented  by  means  of  the  symbols  0,  1,  2,  3,  4,  5,  6, 
7,  8,  9.     But  letters  also  may  be  used  to  stand  for  numbers. 

For  example : 

p  may  stand  for  the  number  of  pounds  in  the  weight  of  a  body  ; 

d  may  stand  for  the  number  of  dollars  in  a  sum  of  money  ; 

1  may  stand  for  the  number  of  units  in  the  length  of  an  object,  and 
the  like. 

2.  The  Use  of  Signs.  The  signs  +,  — ,  =,  X,  -^,  and  V 
have  the  same  meaning  in  algebra  as  in  arithmetic.  But  in 
algebra,  multiplication  is  indicated  also  by  the  absence  of  a  sign 
of  operation.  When  a  sign  is  needed,  the  dot,  • ,  is  often  used  in 
preference  to  the  symbol  x ,  which  is  likely  to  be  mistaken 
for  the  letter  x. 

For  example  : 

a  plus  6  is  written  a  +  6,  just  as  3  plus  2  is  written  3  +  2. 

a  minus  h  is  written  a—  b,  just  as  3  minus  2  is  written  3  —  2. 

a  divided  by  h  is  written  «  -f-  6  or  - ,  just  as  3  divided  by  2  is  written 
q  b 

3  --  2  or  ? . 
2 

2  times  5  is  written  2  x  6  or  2  •  5. 

a  times  b  is  written  ab.  2  times  a  is  written  2  a.  And  2  times  a  plus  b 
times  c  is  written  2  a  -\-  be. 

The  square  root  of  a  is  written  y/a  ;  the  cube  root  of  a,  Va  ;  and  so  on. 

3.  The  use  of  letters  to  represent  numbers  enables  us  to 
write  statements  in  very  brief  form.  This  is  an  important 
feature  of  algebra. 

1 


2  A   HIGH   SCHOaL   ALGEBRA 

For  exaippk  . 

1.  The  length  of  k  lot  diminished  by  ^  of  its  length  is  60  ft. 

Using  I  for  the  number  of  feet  in  the  length  of  the  lot,  this  statement 
may  be  written  :  ?  minus  i  Z  is  60, 

or,  Z  -  it  Z  =  60. 

2.  A  man's  weight  when  increased  by  \  of  itself  is  200  lb. 

Using  w  for  the  number  of  pounds  in  the  man's  weight,  this  statement 
may  be  written:  ^  plus  i  u;  is  200, 

or,'w  +  \w  =  200. 

ORAL  EXERCISES 

1.  If  Z  represents  the  number  of  yards  in  the  length  of  a 
street,  what  stands  for  the  length  of  a  street  75  yd.  longer  ? 

2.  If  w  represents  the  number  of  rods  in  the  width  of  a 
farm,  what  represents  the  width  of  a  farm  20  rd.  narrower  ? 

3.  One  bank  contains  d  dollars  and  another  3  times  as 
many.  How  many  dollars  in  the  second  bank  ?  How  many 
in  both  banks  ? 

4.  There  are  n  pupils  in  a  class  and  the  same  number  in- 
creased by  13  in  another.  How  many  pupils  in  the  second 
class  ?     In  both  classes  ? 

5.  A  merchant  invested  s  dollars  and  lost  -^-^  of  this  the 
first  year.     How  much  had  he  left  ? 

6.   The  line  BC  in  the  fisrure  is  3 

ABC  ^ 

L,^.,^'  times  as  long  as  AB.     If  AB  is  I  units 

I  long,  how  long  is  50?     AC? 

7.  In  the  following  statements  c  stands  for  cost,  s  stands  for 
selling  price,  and  g  for  gain.     Read  each  statement  in  words  : 

1.    s  —  c  =  g.  2.    c-\-g  =  s.  S.    s  —  g  =  c. 

4.  Algebraic  Sjrmbols.  Letters  and  other  characters  used 
as  notations  in  algebra  are  called  algebraic  symbols. 

5.  Algebraic  Expressions.  Any  expression  representing  a 
number  by  use  of  algebraic  symbols  is  called  an  algebraic 
expression. 


LITERAL   NOTATION   AND   ITS   USES  3 

For  example :  3  h  and  2  a  —  6c  +  d  are  algebraic  expressions. 
The  term  literal  expression  is  often  used  to  denote  an  algebraic  expres- 
sion involving  letters. 

6.  Value  of  Algebraic  Expressions.  The  number  represented 
by  an  algebraic  expression  is  called  its  value.  The  value  of  an 
algebraic  expression  is  found  by  substituting  numbers  for  the 
letters. 

Thus,  when  a  =  3,   2  a  =  2  •  3,  or  6. 
Similarly,  when  n  =  5,   2  n  —  1  =  2  •  5  —  1,  or  9. 

ORAL  EXERCISES 

1.  What  is  the  value  of  2  n  when  n  is  1  ?     When  n  is  2  ? 

2.  What  number  is  ^  w  when  n  is  2?  When  n  is  6  ?  When 
nis  10  ? 

3.  State  the  value  of  2  w  when  n  is  ^.  Also  when  n  equals 
each  of  the  following :  f ;  7^ ;  10 ;  .5  ;  1.5 ;  50. 

4.  State  the  value  of  w  +  1  when  n  equals  each  of  the  fol- 
lowing :  1;  2;  6;  5;  i;  .5;  81;  100;  0. 

5.  The  length  (I)  of  a  box  is  twice  its  width  {w).  (I)  is 
(?)w. 

6.  How  many  ounces  are  there  in  5  lb.  ?     In  x  lb.  ? 

7.  A  pair  of  gloves  costs  c  cents.  What  would  the  cost  be 
if  the  price  were  raised  5  cents  ? 

8.  There  were  b  books  on  a  shelf  and  2  were  taken  down. 
How  many  remained  on  the  shelf  ? 

9.  A  person  is  x  years  of  age  now.  How  old  will  he  be  a 
year  hence  ?  5  years  hence  ?  How  old  was  he  3  years  ago  ? 
y  years  ago  ? 

10.  A  merchant  sold  goods  at  8  %  above  cost.  If  c  was  the 
number  of  dollars  in  the  cost,  the  gain  was  8  %  of  c,  or  .08  c. 
What  was  the  selling  price  ? 

11.  Some  goods  costing  x  dollars  were  sold  at  a  gain  of 
150%.     State  the  selling  price. 


4  A  HIGH   SCHOOL  ALGEBRA 

WRITTEN   EXERCISES 

1.  Write  the  sum  of  h  and  c,  using  the  sign  of  addition. 

2.  Indicate  the  subtraction  of  c  from  h  by  using  the  sign 
of  subtraction. 

3.  Write  the  product  of  a  and  h  as  it  is  expressed  in  algebra. 

4.  Write  the  product  of  3  and  h  and  c  as  it  is  expressed  in 
algebra. 

5.  Indicate  that  a  is  to  be  divided  by  h  by  using  the  frac- 
tional form. 

6.  Indicate  that  the  sum  of  a  and  h  is  to  be  divided  by  c. 

7.  Find  the  value  of  2  a  + 1  when  a  equals  each  of  the 
follov^ing:  4;  7;  f;  11|;  15;  .5;  L5;  100;  0. 

Find  the  value  of  a-\-h,  when  a  and  h  indicate  in  turn  the 
following  numbers : 

8.  a  =  2,  6  =  1.        10.    a  =  12,  6  =  9.        12.    a  =  .8,  6  =  .5. 

9.  a  =  14,  5  =  6.      11.   a  =  i,  6  =  i.  13.    a  =  lf,  6  =  |. 
For  each  pair  of  values  of  a  and  h  above,  find  the  correspond- 
ing value  of : 

14.   a-h.  15.   ah.  16.   -•  17.    ^^i^- 

h  ah 

18.  Draw  a  rectangle ;  write  h  for  its  base  and  a  for  its  alti- 
tude. Express  the  area  of  the  rectangle ;  its  perimeter  (sum 
of  its  four  sides). 

19.  A  rectangular  bin  is  a  ft.  long,  h  ft.  wide,  and  c  ft.  deep. 
How  many  cubic  feet  does  it  contain  ? 

20.  What  number  does  100  a  +  10  6  -f  c  represent  when 
a=l,  6  =  2,  c  =  3?     When  a  =  5,  6=4,  c  =  7? 

7.  Tabulation  of  Values.  In  recording  corre- 
sponding values  it  is  convenient  to  use  a  table 
like  that  adjoining. 

The  values  of  a  are  written  in  the  column  under  a, 
and  the  corresponding  values  of  2  a  +  1  are  written  op- 
posite in  the  second  column.  The  table  records,  for 
example,  that  when  a  is  5,  2  a  +  1  is  11 ;  that  is,  2  x  5  -|-  1. 
Verify  the  other  values. 


a 

2a  +  l 

^ 

11 

0 

1 

3 

7 

H 

18 

3.5 

8 

LITERAL  NOTATION  AND  ITS  USES 


WRITTEN   EXERCISES 


Copy  the  following  tables  and  supply  the  numbers  to  fill  the 
blanks : 


L.    n 

3n 

0 

1 

4 

5 

H 

¥ 

4.     n 

n-1 

1 

2 

7 

6* 

18 

25 

V 

10  V 

.1 

1.2 

1.5 

6.3 

40.4 

.05 

2.    n 

271-3 

2 

2i 

3 

f 

1.5 

5.     a 

ia  +  1 

2 

12 

25 

1.8 

4.6 

w 

|w 

0 

1 

24 

64 

100 

3.     a 

b 

a  +  b 

4 

5 

(     ) 

6i 

2i 

(     ) 

1.8 

9.2 

(     ) 

2.5 

8.3 

(     ) 

V 

f 

v^ 

m 

H 

(     ) 

1.8 

.7 

(     ) 

4 

3.8 

(     ) 

12 

7i 

(     ) 

9.     I 

6 

< 

;6« 

6 

8 

3 

(  ) 

4 

5 

6 

(  ) 

4 

i 

i 

(  ) 

* 

1 

.8 

(  ) 

REVIEW 


ORAL  EXERCISES 

1.  A  man  invests  d  dollars  in  real  estate,  and 
much  in  government  bonds.  How  much  does 
altogether  ? 

2.  In    the    figure    BC    is    i 
twice  as  long  as  AB,  and  CD 


5  times  as 
he    invest 


B 


is  three  times  as  long  as  AB. 
AB,  what  denotes  the  length 
entire  line? 


If  I  denotes  the  length   of 
of  BC?     Of  CD?     Of  the 


6  A   HIGH   SCHOOL   ALGEBRA 

Find  the  value  of  A-\-B,  when  A  and  B  have  the  following 
values  : 

3.    ^  =  7,5  =  12.  5.   A  =  Sx,B  =  4:X. 

6.   A  =  5y,B  =  Ty. 

WRITTEN   EXERCISES 
1.    Copy  and  fill  the  blanks : 


4.   A  =  i,B  =  l. 


(1) 

(2) 

(3) 

(4) 

(5) 

Given 
Find 

n  = 
t  = 

371-1  = 

7l+i  = 

[        2nt= 

6 
4 

1 

i 

t 

1 

100 
10 



1.3 

20. 

2.  The  length  of  one  line  is  x  and  that  of  another  is  y. 
How  long  is  the  line  formed  by  placing  them  end  to  end  ? 

3.  Line  x  is  longer  than  line   y.     Express   the   difference 
between  their  lengths. 

4.  What  number  does  -  represent  when  a  =  29,  and  6  =  68  ? 

b 

When  a  =  35,  and  6  =  70?    Whena  =  127,  6  =  210? 


SUMMARY 

The  following  questions  summarize  the  definitions  and  pro- 
cesses treated  in  this  chapter : 

1.  What  symbols  may  stand  for  numbers  ?  Sec.  1. 

2.  Name  the  different  signs  of  operation  used  in  algebra. 

Sec.  2. 

3.  In  what  ways  may  multiplication  be  indicated  in  algebra  ? 

Sec.  2. 

4.  What  is  an  algebraic  expression?  Sec.  5. 

5.  What  is  meant  by  the  value  of  an  algebraic  expression  ? 

Sec.  6. 


LITERAL   NOTATION   AND  ITS  USES  7 

HISTORICAL  NOTE 

If  algebra  appears  more  abstract  than  other  school  studies,  if  it  seems 
less  real  than  arithmetic,  this  is  largely  on  account  of  its  use  of  letters  and 
other  symbols  of  abbreviation.  We  have  learned  to  express  "any  num- 
ber "  by  a  single  letter,  like  a  or  x,  to  abbreviate  the  "square  of  a 
number ' '  by  cfi  or  ic^,  and  to  express  operations  with  these  numbers  by 
signs  like  +,  — ,  x,  =;  all  of  this  seems  artificial  and  abstract,  but 
this  is  one  of  the  chief  sources  of  algebra's  power.  With  these  symbols 
we  can  easily  find  in  a  few  minutes  results  which  the  ancients  sought  in 
vain.  Thus,  through  lack  of  a  suitable  notation,  the  Greeks  made  slow 
progress  in  studying  the  processes  with  numbers  as  well  as  those  phases 
of  geometry  that  require  calculation,  like  the  measuring  of  circles  and 
polygons  ;  and  it  was  not  until  300  a.d.  that  Diophantos,  who  taught  in 
the  University  of  Alexandria,  Egypt,  developed  algebra  into  a  science. 

History  tells  little  of  the  life  of  Diophantos.  His  birthplace,  his  par- 
entage, his  early  education,  and  the  steps  which  led  to  his  great  achieve- 
ments are  unknown.  He  died  in  330  a.d.  ,  and  his  age  at  death  is  shoTwn 
to  have  been  84  years  by  the  following  epitaph  :  "  Diophantos  passed  one- 
sixth  of  his  life  in  childhood,  one-twelfth  in  youth,  and  one-seventh  more 
as  a  bachelor.  Five  years  after  his  marriage  was  born  a  son,  who  died 
four  years  before  his  father  at  half  his  father's  age."  It  may  seem  strange 
to  resort  to  this  indirect  means  of  telling  the  age  of  a  famous  man,  but  it 
was  the  custom  of  the  ancients  to  inclose  in  the  tombs  of  great  men  some 
possession  or  record  associated  with  their  lives ;  and  this  is  one  of  the 
chief  sources  of  early  history. 

Diophantos  vra,s  the  first  to  use  abbreviations  and  symbols  in  algebra. 
On  this  account,  and  because  of  his  simple  solution  of  many  problems 
then  thought  difficult,  he  has  been  called  the  Father  of  Algebra. 


CHAPTER   II 
DEFINITIONS  OF  ELEMENTARY  TERMS 

8.  Product  and  Factors.  In  algebra,  as  in  arithmetic,  the 
result  of  multiplication  is  called  a  product,  and  the  numbers 
multiplied  are  called  the  factors  of  the  product. 

9.  A  number  may  be  the  product  of  different  sets  of  factors. 

10.  Literal  and  Numerical  Factors.  Factors  expressed  by 
letters  are  called  literal  factors ;  factors  expressed  by  numerals 
are  called  numerical  factors. 

For  example : 

10  is  the  product  of  the  factors  2  and  5. 

2  ax  is  the  product  of  the  factors  2,  a,  and  x,  in  which 

2  is  a  numerical  factor,  but  a  and  x  are  literal  factors. 

In  a  product  it  is  customary  to  put  the  numerical  factor  (if 
any)  first,  and  the  literal  factors  in  alphabetical  order. 

11.  Commutative  Law  of  Multiplication.  The  product  is  the 
same  in  whatever  order  the  factors  are  taken. 

For  example  :  S  ■  ab  '  x  gives  the  same  product  as  ab  -B  -  x  and  the 
same  as  abx  -3. 

12.  Unity  is  a  factor  of  every  number,  but  it  is  not  ordi- 
narily mentioned  in  giving  lists  of  factors. 

Thus,  a  set  of  factors  of  abc  are  1,  a,  6,  and  c ;  but  1  is  usually  not 
mentioned. 

ORAL  EXERCISES 
Name  a  set  of  factors  for  each  of  the  following  products : 

1.  10.  3.  prt.  5.   |a6.  7.   gt. 

2.  3  a.  4.   Ir,  6.   mv,  8.   i  sf. 

8 


DEFINlTIOxXS   OF   ELEMENTARY   TERMS  9 

Name  three  sets  of  factors  for  each  of  the  following  and 
state  which  factors  are  literal  and  which  are  numerical: 
9.    40.  11.   2oab.  13.  prt.  15.    75 pq. 

10.    18  a.  12.    20  hr.  14.   25  my.  16.   abed. 

13.  Power  and  Base.  A  product  formed  by  using  the  same 
number  one  or  more  times  as  a  factor  is  called  a  power  of  the 
repeated  factor.     The  repeated  factor  is  called  the  base. 

14.  Exponent.  When  a  factor  is  to  be  repeated,  it  is  usual 
to  write  the  factor  only  once  and  place  a  small  number  above 
and  to  the  right  to  show  how  many  times  the  number  is  to  be 
used  as  a  factor.     The  small  number  is  called  an  exponent. 

Thus,  2  •  2  is  the  second  power  (or  square)  of  2,  and  is  written  2^ ; 
a  .  a  •  a  is, the  third  power  (or  cube)  of  a,  and  is  written  aK  Similarly, 
3  aabbb  is  written  3  a^b^. 

15.  A  number  without  an  exponent  is  understood  to  have 
the  exponent  1,  since  the  number  is  used  once  as  a  factor. 

Thus,  5  =  51 ;  «=?«!;  7  xyz^  =  7^  x^y'^z^. 

16.  The  factors  of  numbers  can  be  conveniently  grouped  by 
the  use  of  exponents. 

For  example  : 

12  =  2  .  2  •  3  =  22  .  3.  144  =  12  .  12  =  122. 

225  =  3  .  3  •  5  .  5  =  32  .  52.  600  =  2^  .  3  •  52. 

17.  Prime  Numbers.  As  in  arithmetic,  so  in  algebra,  an 
integer  whose  only  integral   factors  are  itself   and   unity  is 

*  called  a  prime  number. 

18.  Prime  Factors.  Prime  numbers  occurring  as  factors  are 
called  prime  factors. 

A  number  has  only  one  set  of  prime  factors. 

ORAL  EXERCISES 

Name  the  exponents  and  tell  what  each  means : 

1.    2aV.  2.   3a^.  3.    o  xf.  4.    a^M.  5.    2a^y. 


10  A  HIGH   SCHOOL   ALGEBRA 

WRITTEN   EXERCISES 
Regarding  each  letter  as  prime^  indicate  the  prime  factors 
of: 

1.  18.  3.    96.  5.   640.  7.   360.  9.    1225. 

2.  75  a.        4.    12i«2.       6.    24.  ab.        8.    17  a%       10.   38  orV- 

19.  Coefficient.  Any  factor  in  a  product  is  called  the  coeffi- 
cient of  the  rest  of  the  product. 

Thus,  in  the  product  3  axy,  3  is  the  coeflBcient  of  axy,  S  a  is  the  coeffi- 
cient of  xy^  3  ax  is  the  coefficient  of  y,  3  ay  is  the  coefficient  of  x,  and  the 
like. 

20.  Numerical  Coefficient.  A  coefficient  expressed  in  nu- 
merals is  called  a  numerical  coefficient. 

Thus,  3  is  the  numerical  coefficient  in  3  x,  and  |  is  the  numerical  coeffi- 
cient in  ^xy. 

The  term  "  coefficient,"  used  with  no  other  indication,  means 
the  numerical  coefficient. 

21.  In  any  product  whose  numerical  coefficient  is  not  ex- 
pressed, the  coefficient  1  is  understood. 

Thus,  ah,  abx,  bc^y,  are  the  same  as  lab,  I  abx,  1  bc'^y ;  and  the  nu- 
merical coefficient  in  each  is  1. 

ORAL  EXERCISES 

1.  In  6  ab  name  the  coefficient  of  ab.     The  coefficient  of  b. 

2.  In  I  axy  name  the  coefficient  of  xy.     Of  axy.     Of  y. 

Name  the  numerical  coefficient  in  each  of  the  following : 

3.  4.  5.  6.  7.  8.  9.  10.  11. 
2  X.      3y.      I- ax.      by.      .5  cz.      ^my.      \gf.      xyz.      ^mr'K 

22.  Order  of  Operations.  In  an  expression  containing  a 
series  of  operations,  multiplications  and  divisions  are  to  be 
performed  before  additions  and  subtractions,  unless  otherwise 
indicated. 

Thus,  4  4  5-3  means  4  +  15,  or  19. 

Similarly,  3-H8--2-5  =  3-f-4-5=7--5  =  2. 


DEFINITIONS  OF  ELEMENTARY  TERMS  11 

ORAL  EXERCISES 
Perform  the  operations  indicated : 


1. 

5-4-3. 

5. 

40-34-2. 

9. 

18-^2  +  3.6. 

2. 

9+2.6. 

6. 

52-^26-1. 

10. 

3  -1-  4  .  5  - 13. 

3. 

14-8-4. 

7. 

10  .  3  -  3  .  8. 

11. 

a  +  2a-j-a  —  a. 

4. 

15  _  12  -  4. 

8. 

3.9_15^5. 

12. 

b  '  a-\-a  '  b. 

23.  The  Use  of  the  Parenthesis.  In  a  series  of  operations 
the  parenthesis  may  be  used  to  indicate  that  certain  additions 
and  subtractions  are  to  be  performed  first. 

For  example  : 

6  +  4-3  means  add  6  to  4  times  3,  obtaining  18.  But  (6  +  4)  •  3 
means  add  6  and  4  and  multiply  the  result  by  3,  obtaining  30.  In  other 
words,  what  is  in  the  parenthesis  is  to  be  treated  as  a  single  number. 

(4  -j_  16)  -^  2  means  20  -^  2,  or  10,  and  not  4+  — ,  or  12. 

8  —  (12  —  7)  means  that  7  is  first  to  be  subtracted  from  12  and  then 
the  result  subtracted  from  8.     That  is,  8  —  (12  —  7)  =  8  --  5  =  3. 
14  a  —  (7  a  +  3  a)  =  14  a  -  10  a  =  4  a. 

24.  When  a  number  symbol  is  placed  before  or  after  a  pa- 
renthesis with  no  intervening  sign,  multiplication  is  indicated. 

For  example  ; 

2  a{c  +  2  c)  means  2  a  •  3  c,  or  6  ac. 

(4  a;  +  5  x)i  x  means  9  x  •  4  x  or  36  a:^. 

(a  +  4  a)  (8  6  —  5  &)  means  5  a  •  3  &,  or  15  ab. 

3(6  +  2  c)  — c  means  3  6  +  6  c  —  c,  the  multiplication  by  3  being  per- 
formed before  c  is  subtracted. 

5(100  —  x)  +  25  means  500  —  5  x  +  25,  or  525  —  5  x,  the  multiplication 
by  5  being  performed  before  25  is  added. 

ORAL  EXERCISES 

Perform  the  operations  indicated  : 

1.  (i5_6)-f.3.  6.  (24 +  6) -(8 -3). 

2.  7(25  +  5).  7.  (2a+-a)--3. 

3.  (18-12)h-6.  8.  (2a-\-3a)b. 

4;   5(6+-5-9).  '      9.  (2a  +  3a)--(36  +  26). 

5.    (2  +  5).  (5-3).  10.  a-(6  +  26-36). 


12  A  HIGH  SCHOOL  ALGEBRA 

11.  ab(Sc-2c)-\-d.  15.  19-(4  +  7). 

12.  m(m-\-5m)  —  2m.  16.  8  a  — (7  a  — 3  a). 

13.  (2a  +  a)  .(3c-c).  17.  43  a?  +  (28  a;- 8  a;). 

14.  x(ox—2x)—y(y-\-4:y).  18.  S6y  —  (10  y —  4:y) +  10y. 

25.  S3nnbols  of  Grouping.  The  parenthesis  is  used  to  indi- 
cate  that  the  number  symbols  grouped  within  it  are  to  be  taken 
as  a  single  number.  Other  symbols  of  grouping  are  the  brace, 
J  j,  the  bracket,  [  ],  the  bar,  ;  these  have  the  same  mean- 
ing as  the  parenthesis.  The  bar  of  the  fraction  may  also  be  a 
symbol  of  grouping. 

Thus,  in  ^  "*"    ,  the  bar  groups  a  +  6  into  one  number,  and  c  +  (Z  into 
c  -{-  d 

one  number.    The  fraction  means  (a  +  6)  -^  {c  +  d). 

26.  Monomials.  A  monomial  is  an  algebraic  expression 
within  which  no  operation  of  addition  or  subtraction  is  indi- 
cated, unless  within  a  symbol  of  grouping. 

Thus,  a,  a6,  a  -^  2  &,  ^,  7  (a  +  &),  ^^~'^,  are  monomials, 
c'^  X 

27.  Polynomials.  An  algebraic  expression  consisting  of  two 
or  more  monomials  connected  by  the  sign  -f-  or  —  is  called  a 
polynomial.  The  monomials  are  called  the  terms  of  the  poly- 
nomial. 

Thus,  a  +  56  +  c  +  -isa  polynomial  whose  terms  are  a,  6  b,  c,  and  -  • 

28.  Binomials.  A  polynomial  of  two  terms  is  called  a  bi- 
nomial. 

Thus,  a  +  b,  b^  -  c,  xy  +  m,  S  b'  -  a,  ^-  —,  Sire  binomials. 

29.  Trinomials.  A  polynomial  of  three  terms  is  called  a 
trinomial. 

1  Thus,  a  +  6  +  c,  a  +  2  6  -  c,  ^  -  a6  -}-  3  c,  are  trinomials. 

y 

30.  Compound  Terms.  Expressions  are  sometimes  grouped 
into  compound  terms. 

Thus,  Sa  —  2b  +  c  +  d  may  be  grouped  into  the  trinomial  3  a  —  2  6 
-f  (c  +  c?),  in  which  (c  +  d)  is  a  compound  term. 


DEFINITIONS   OF   ELEMENTARY   TERMS  IS 

ORAL  EXERCISES 

Name  the  monomials  in  the  following  list ;  the  binomials ; 
the  trinomials : 

1.  a  +  b  —  c.  5.   gt^-\-a,  9.   Sa^  +  a^/^. 

2.  4a^  +  7.  6.   mv\  10.    2  7ir. 

3.  a-f-6  +  c  +  d.       7.    a  +  5  6  — c.  11.  prt. 

4.  I^il  8.   a;  — 2/H-2  +  W.         12.   2g  —  5-\-x^. 

13.  What  is  the  coefficient  of  t^  in  Exercise  4  ? 

14.  What  is  the  coefficient  of  v^  in  Exercise  6  ? 

15.  What  is  the  numerical  coefficient  in  Exercise  4? 

16.  Name  the  numerical  coefficients  in  Exercise  7. 

17.  Name  the  coefficient  of  7i  in  Exercise  10. 

18.  Read  the  numerical  coefficients  in  Exercise  8. 

*  WRITTEN   EXERCISES 

1.  Write  three  monomials. 

2.  Write  three  binomials.     Three  trinomials. 

Rewrite  these  expressions,  using  exponents  where  possible : 

3.  a  4-  bb.  6.    ccc  —  bb.  9.    15  mvvq. 

4.  2  aa  +  b.  7.   3  aayyy.  10.    16  xxy  —  cd. 

5.  aa-bbb.  8.   2  -  2  -  2  bbb.  11.   100  aa66  -  sss. 

Using  a  =  1,  6  =  2,  and  c  =  3,  find  the  value  of  each  of  the 
following  polynomials ; 

12.  5a-\-9b.  21.  3a-7  6  +  llc. 

13.  10  a- 56.  "        22.  61  6- 2  c- 20 a. 

14.  2a  +  6-c.  23.  ia  +  |c-i&. 

15.  3a  +  15  6.  24.  a'-{-b\ 

16.  2a  +  36-f3c.  25.  b^-aK 

17.  .9a  +  .36-.lc.  26.  ac^'-i-Sb^ 

18.  96  +  2a-c.  27.  7  ab- -  (^. 

19.  2a +3  6 -2c.  28.  c^-b\ 

20.  3a +  7  5+11  c.  29.  c^  —  1 -f  5  d^. 


14  A  HIGH   SCHOOL   ALGEBRA 

31.  Uses  of  Monomials.  Monomials  have  various  uses.  For 
example : 

1.  They  are  used  as  formulas  in  business  arithmetic. 

Thus  : 

hr  is  often  used  as  a  short  way  of  stating  base  times  rate  in  percentage. 

prt  is  often  used  as  a  short  way  of  stating  principal  times  rate  times 
time  in  interest. 

When  any  particular  value  of  r  is  substituted  in  the  above  formulas  it 
must  be  expressed  decimally.    Thus,  if  r  is  5%,  it  must  be  used  as  .05,  or  y^^. 

2.  TJiey  are  used  as  formulas  of  measurement. 

Thus: 

ab  is  often  used  as  a  short  way  of  stating  altitude  times  base  in  finding 
areas. 

abc  is  a  short  way  of  stating  length  times  breadth  times  thickness,  in 
finding  volumes  of  rectangular  solids,  where  a,  6,  and  c  are  the  edges. 

w  (read  "  pT ")  is  used  to  denote  the  number  by  which  the  length  of  the 
diameter  of  a  circle  must  be  multiplied  to  produce  the  circumference. 

The  value  of  tt  is  approximately  3.1416.  Letting  c  =  circumference, 
and  d  =  diameter,  we  have  c  =  ird  =  3.1416  d. 

The  word  circumference,  as  used  above,  means  the  distance  around  the 
circle  or  the  length  of  the  curve.  The  conception  of  a  circle  as  a  curve 
is  the  one  used  in  advanced  mathematics,  in  other  sciences,  and  in  com- 
mon parlance. 

3.  They  are  used  to  express  laws  of  physics. 

Thus,  vt  is  often  used  as  a  short  way  of  stating  product  of  velocity  and 
time  in  finding  distance. 

WRITTEN   EXERCISES 

1.  Percentage  =  hr.     Find  the  percentage  when  b  =  400  and 

30 
r  =  30  % .     In  substituting  r  use  ——  • 

2.  Kate  =  ^-     Find  the  rate  when  p~15  and  b  =  750. 

b 

3.  Principal=--  Find  the  principal  when  i=$500,r=5%, 
and«  =  10yr.      ^'^ 

4.  Interest  =pr^.  Find  the  interest  when  jp=  $100,  r=5%, 
and  t  =  5  yr. 


DEFINITIONS  OF   ELEMENTARY  TERMS 


15 


d 

ird 

2  in. 

6.2832  in. 

3  in. 

(          ) 

iin. 

(          ) 

1.5  in. 

(         ) 

10  ft. 

(          ) 

5.  Discount  =  h\     Find  the  discount  when  I  =  $  820  and 
r  =  12i%. 

6.  Rate  of  discount  =  - •  Find  the  rate  when  d=$S5  and 
/  =  $875.  ^ 

7.  The  area  of  a  rectangle  =  ab.  Find  the  area  when  a  = 
20  in.  and  &  =  171  in. 

8.  What  is  the  area  of  a  rectangle  when  a  =  4  a;  inches  and 
b  =  Sx  inches  ? 

9.  Copy  the  following  table  and  fill  out  the  blanks,  using  the 
value  of  TT  mentioned  in  Sec.  31.     Answer 
from  your  table : 

(a)  What  is  the  circumference  of  a  cir- 
cle whose  diameter  (d)  is  3  in.  ? 

(b)  Of  one  whose  diameter  is  1.5  in.  ? 

(c)  Of  one  whose   diameter   is   ^  in.  ? 
10  ft.  ? 

10.  The  distance  (d)  traveled  by  a  body  in  time  (t)  moving 
with  velocity  (v)  is  vt.     Copy  the  following 
table  and  fill  out  the  blanks.    Answer  from 
your  table : 

(a)  How  far  will  a  train  moving  30  ft. 
per  second  go  in  2  sec.  ? 

(b)  How  far  will  a  train  moving  50  mi. 
per  hour  travel  in  2  hr.  ? 

(c)  How  far  will  a  bullet  traveling  400  ft.  per  second 
in  5.5  sec.  ? 

11.  The  number  of  square  units  in  the  area  of  a  triangle  is 
^  of  the  product  of  the  numbers  of  linear  units  in  its  altitude 
(a)  and  base  (6).     Copy  this  table  and  fill  out  the  blanks. 

Answer  these  questions  from  your  table : 

(a)  What  is  the  area  of  a  triangle  of  alti- 
tude 10.5  in.  and  base  8  in.  ? 

(b)  What  is  the  area  of  a  triangle  of  alti- 
tude 70  ft.  and  base  9  ft.  ? 

(c)  What  is  the  area  of  a  triangle  of  alti- 
tude 3.3  yd.  and  base  5  yd.  ? 


V 

t 

vt  =  d 

30 

2 

60 

50 

2 

(     ) 

36 

H 

(    ) 

400 

5.5 

(     ) 

150 

17 

(    ) 

go 


a 

b 

^ab 

6 

5 

15 

10.5 

8 

(    ) 

70 

9 

(    ) 

3.3 

5 

(    ) 

9 

1.7 

(    ) 

16.  A  HIGH  SCHOOL  ALGEBRA 

32.   Uses  of  Polynomials.    Polynomials,  like  monomials,  have 
various  uses  as  formulas. 
For  example  : 

1  —  Ir  may  stand  for  list  price  —  discount^  or  net  price. 

c  +  re  may  stand  for  cost  +  rate  of  gain  times  the  cost,  or  the  selling 
price. 

2  a  4-  2  &  may  stand  for  the  perimeter  of  a  rectangle  of  sides  a  and  b. 

2  ab  +2  ac  -\-  2bc  may  stand  for  the  surface  of  a  rectangular  solid  of 
edges  a,  6,  c. 

ORAL  EXERCISES 

1.  What  is  the  value  of  I  -  rl  when  Z  =  $  100  and  r  =  5  %  ? 
What  is  the  net  price  of  goods  listed  at  $  100  and  bought  at  a 
discount  of  5  %  ? 

2.  What  is  the  value  of  c  -f-  cr,  when  c  =  $  200  and  r  =  10  %  ? 
W^hat  is  the  selling  price  of  goods  which  cost  $200  and  are 
sold  at  a  gain  of  10  %  ? 

a=  2  3.    When  a  =  3,  6  =  4,  c  =  5y  cZ  =  6,  what  is  the 

(f     value  oia-\-h-\-c-\-d? 

^        4.   What  is  the  value  of  a -\- b -\- c -{- d  in  the 
figure  ? 

5.  What  is  the  value  of  2  a  +  2  6  when  a  =  3,  6  =  5  ?  What 
is  the  perimeter  of  a  rectangle  whose  sides  are  3  yd.  and 
5  yd.? 

WRITTEN   EXERCISES 

1.  When  a  =  10,  5  =  15,  c  =  24,  find  the  value  of  2  ab  -\-2ac 
+  2  be.  What  is  the  area  of  the  surface  of  a  rectangular  prism 
whose  edges  are  20  in.,  15  in.,  50  in.? 

2.  Find  the  value  of  2  a6  +  2  6c  +  2  ac,  when  a  =  20,  6  =  25, 
c  =  50. 

3.  Find  the  value  of  a^  +  6^  when  a  =  13,  6  =  210 ;  also  when 
a  =  75,  5  =  100. 

4.  Find  the  value  of  a^  -f  6^  4.  c2,  when  a=  35,  6  =  20,  c  =  65; 
also  when  a  =  100,  6  =  75,  c  =  150. 

5.  Find  the  value  of  ut  +  i  af^,  when  u  =  1500,  a  =  200, 
^  =  10. 


DEFINITIONS  OF  ELEMENTARY  TERMS  17 

33.  Degree  of  a  Monomial.  The  degree  of  a  monomial  is 
the  sum  of  the  exponents  of  its  literal  factors. 

Thus  :  a^  is  of  the  second  degree. 

3  ah  is  of  the  second  degree. 
2  a%  is  of  the  fourth  degree. 

But  the  degree  is  often  expressed  with  respect  to  some  letter 
or  letters. 

Thus,  3  ax^-y^  is  of  the  first  degree  with  respect  to  a,  of  the  second  de- 
gree with  respect  to  aj,  of  the  third  degree  with  respect  to  ?/,  and  of  the 
fifth  degree  with  respect  to  x  and  y. 

34.  Degree  of  a  Polynomial.  The  degree  of  a  polynomial  is 
that  of  its  term  of  highest  degree ;  its  degree  with  respect  to  a 
letter  is  the  highest  degree  of  that  letter  in  the  polynomial. 

Thus,  a%2  —  5  6y  +  xy^z  is  of  the  sixth  degree  ;  it  is  of  the  third  degree 
in  a,  the  first  in  h.  and  in  0,  the  second  in  x,  the  fourth  in  y,  and  the  sixth 
In  X,  y,  and  z. 

Note.  It  is  not  necessary  in  elementary  algebra  to  define  the  degree 
of  expressions  containing  radicals  or  fractions. 


ORAL  EXERCISES 

State  the  degree  of  each  of  the  following  monomials : 

1.  a^h.  4.    3  o^x.  7.   5  mn^.  10.    a^yz^. 

2.  ax.  5.   a^h.  8.    ^  xyz.  11.    \mv. 

3.  2  ax.  6.   4a26l  9.   ^  x^bz.  12.   ^gt^. 

13.  State  the  degree  of  the  expressions  in  Exercises  1-6 
with  respect  to  a, ;  in  Exercises  8-10  with  respect  to  x. 

14.  State  the  degree  of  the  expressions  in  Exercises  4-9 
with  respect  to  each  letter  involved. 

State  the  degree  of  each  polynomial;  also  its  degree  with 
respect  to  each  letter : 

15.  ah"  +  h.  18.   ahc -\- (^ -[■  bed. 

16.  a-b -^  a^G -{•  d,  19.   S  ax  +  Z  o^y -\- f. 

17.  3aj2  +  2aj  +  l.     "  20.   ^m^  +  ii^  +  ^pq. 


18  A  HIGH   SCHOOL  ALGEBRA 

REVIEW 
ORAL  EXERCISES 

State  the  product  of  each  set  of  factors : 
1.   8,  6,  a.         2.   h,  X,  3.         3.   a,  y,  3,  4.         4.   2,  a;,  a,  a;. 

Name  three  sets  of  factors  for  each  of  the  following : 
5.   24  mx.  6.   ax^y.  7.    ^  gt\  8.    21  a5c. 

In   each  of  the  following  name    (1)  the  coefficient  of  aj; 
(2)  the  numerical  coefficient: 

9.   4  ax.      10.   12  bx.       11.    aca;.       12.    5  mxy.       13.   12  c^x. 

State  the  value  of : 

14.    2\  15.    72.  16.    53.         17.    32. 2^       18.    23-52. 

From  the  following  list  select  by  number  the  binomials ;  the 
trinomials ;  the  monomials : 

19.  ax^.  23.   3a2/2-4a:  +  l. 

20.  a  —  x.  24.   2a6  4-7a^  — 5a;2. 

21.  2a^-5a.  25.   a''-2ab-hb\ 

22.  6a-{-7xy.  '  26.   a;' +  3  a.-^  +  3  a;  4- 1. 

27.  State  the  degree  of  each  expression  in  Exercises  19-26 
with  respect  to  x.     With  respect  to  a. 

28.  The  sides  of  a  triangle  are  3  a,  2  b,  5  c.     What  is  its 
perimeter  ?     What  kind  of  polynomial  is  this  ? 

29.  What    is    the   value    of    c -{- cr    when    c=$500    and 
r  =  20%? 

30.  What  is  the  value  of  a;^  -f  5  a  when  a;  =  3  and  a  =  2  ? 

State  the  results  of  the  indicated  operations  : 

31.  39-^13  +  1.      35.    (8  +  5)2.  39.   8a;H-5a;-a;. 

32.  8  +  5.2.  36.    7(8-6).  40.    8  -  (5  +  2). 

33.  7.8-6.  37.    8-5  +  2.  41.    7a-(3a  +  2a). 

34.  39-r-(13  +  l).   38.    7a-3a-{-2a.    42.    8  a;  +  (5  a;  -  a;). 


DEFINITIONS  OF  ELEMENTARY   TERMS  19 

WRITTEN   EXERCISES 

Indicate  the  prime  factors  of  the  following,  using  exponents 
where  possible : 

1.   88.  2.   144.  3.   200.  4.   525. 

Factor  so  that  one  factor  of  each  is  a  power  of  10 : 
6.    1000.  6.   900.  7.    23,000.  8.   3,000,000. 

Find  the  value  of  each  of  the  following  if  a  =  2^  6  =  3,  ic  =  1 : 

9.    ^ax  +  X.     10.   g"f-l.       11.    ^llH^'.       12.    «'  +  4 


26  x-\-\  6  +  2  a; 

13.  The  area  of  a  triangle  is  \  ah.     What  is  the  area  of  a 
triangle  in  which  a  =  3  n  feet  and  6  =  14  w  feet  ? 

14.  Find  the  value  of  a^  +  2  a6  +  6^  when  a  =  7,  6  =  3. 

15.  Find  the  value  of  a^  —  3  o}x  +  3  ax^  when  a  =  7,  »=  2. 

SUMMARY 
The  following  questions  summarize  the  definitions  and  pro- 
cesses treated  in  this  chapter  : 

1.  'Dq^wq  product,    Al^o  factor.  Sec.  8. 

2.  What  is  a  numerical  factor  ?  A  literal  factor  ?  Sec.  10. 

3.  Explain  the  order  of  the  factors  in  a  product.  Sec.  10. 

4.  State  the  Commutative  Law  of  Multiplication.  Sec.  11. 

5.  J) Q^ne  exponent ;  also  6ase;  dl^o  power.         Sees.  13,  14. 

6.  Define  coefficient;  also  numerical  coefficient.  Sees.  19,  20. 

7.  Name  the  usual  symbols  of  grouping.  Sees.  23,  25. 

8.  What  process  is  indicated  by  a  number  standing  next  to 
a  parenthesis  with  no  intervening  sign  ?  Sec.  24. 

9.  Define  monommZ;  2i\^o  polynomial.  Sees.  26,  27. 

10.  Define  binomial;  also  trinomial.  Sees.  28,  29. 

11.  What  is  a  compouyid  term?  Sec.  30. 

12.  How  is  the  degree  of  a  monomial  found  ?     Also  the 
degree  with  respect  to  a  given  letter  ?  Sec.  33. 

13.  What  is  the  degree  of  a  polynomial  ?  Sec.  34. 


20  A   HIGH   SCHOOL   ALGEBRA 

HISTORICAL  NOTE 

To  understand  how  the  symbols  of  algebra  came  to  be  what  they  are, 
we  may  again  refer  to  Diophantos,  for  in  his  work,  Arithmetical  he 
made  the  first  approach  to  algebraic  notation  by  using  certain  signs  and 
abbreviations.  Since  these  characters  are  formed  from  Greek  letters  and 
words,  it  would  be  confusing  to  illustrate  more  than  a  few  of  them,  and 
we  will  confine  ourselves  to  showing  how  Diophantos  would  have  written 
the  polynomial,  3  x^  +  a;  —  2.  His  coefficients  were  placed  after  the  literal 
factors  ;  hence,  3  x^  would  be  x^  3,  and  x  would  be  x  1.  We  must  also 
remember  that  the  coefficients  were  expressed  by  the  Greek  numerals : 
a  for  1,  ^  for  2,  7  for  3,  and  so  on.  For  x  he  used  the  symbol  s,  prob- 
ably a  contraction  of  the  first  two  letters  of  the  Greek  word  meaning 
*' unknown  number."  Diophantos  did  not  know  of  exponents,  but  for  x^ 
he  used  5",  an  abbreviation  of  the  word  meaning  "  square."  His  sign  for 
minus  was  //i,  derived  from  the  letters  of  the  word  meaning  "minus." 
He  had  no  sign  for  plus,  but  placed  numbers  next  to  each  other  to  indi- 
cate addition.  Thus,  3  x^  +  x  —  2,  if  written  by  Diophantos, 
would  be  arranged  thus,  x^  3  x  1  —  2 
and  look  like  5"  7  s  a  /71  ^ 

The  notation  of  algebra,  as  we  now  use  it,  was  developed  about  the 
sixteenth  century.  By  that  time  the  processes  came  to  have  signs  to 
indicate  them  ;  for  example,  addition  and  subtraction  were  denoted 
by  p,  w,  probably  abbreviations  for  the  words  "plus"  and  "minus," 
and  these  were  finally  supplanted  by  4-  and  — .  The  latter  forms  may 
have  resulted  from  the  rapid  writing  of  the  letters  p  and  m,  but  a  more 
likely  origin  is  found  in  the  commercial  arithmetic  of  that  time.  In 
recording  weights  of  merchandise  as  marked  in  the  warerooms,  the 
sign  +  was  used  to  denote  overweight,  and  the  sign  —  to  denote 
short  weight.  Thus,  if  a  bale  supposed  to  weigh  100  pounds  weighed  103 
pounds,  it  was  recorded  as  100  +  3,  or  if  it  weighed  97  pounds,  it  was 
recorded  as  100  —  3.  From  this  practice  the  signs  +  and  —  probably 
came  to  be  generally  used  as  signs  of  addition  and  subtraction.  The  sign 
X  was  first  used  by  the  English  mathematician  Oughtred  in  1631,  and 
the  simpler  sign,  .  ,  for  multiplication  was  used  by  the  famous  French 
mathematician,  Ren6  Descartes.  For  the  square  of  a  he  used  aa,  which 
later  became  a^.  The  symbol  for  square  root  became  B,  the  initial  letter 
of  the  Latin  word  radix  ("  root ")  ;  this  was  afterward  changed  to  y/  by 
the  German  algebraist,  Stifel  (1554).  (The  abbreviation  B,  now  used  in 
physicians'  prescriptions,  denotes  similarly  the  first  letter  of  the  Latin 
word  recipe  —  "take.")  The  sign  of  equality  (  =  )  was  first  used  by 
Robert  Recorde,  an  Englishman,  1542.  His  explanation  of  this  sign  was 
that  "  Noe.  2.  thynges  can  be  moare  equalle  "  than  these  parallel  lines. 


CHAPTER  III 


THE   EQUATION 
35.   Preparatory. 

1.  What  weight,  w,  will  balance  the  package  of  rice  in  the 
figure  ?    What  must  w  be  to  balance  two 
such  packages  ? 

2.  If  half  of  the  rice  be  taken  from 
the  package,  what  must  w  be  ? 

3.  What  must  w  be  to  balance  a  16- 
ounce  package  and  an  8-ounce  package  ? 

4.  What  must  ic  be  to  balance  the  pans  in  each  of  the  fol- 
lowing figures  ? 


Fig.  1. 


FiCx.  2. 


Fig.  3. 


Fig.  4. 


5.  In  Pig.  2  the  fact  that  the  weights  balance  is  expressed 
by  w  H-  3  =  7.  Express  the  condition  that  the  weights  balance 
in  Fig.  3.     In  Pig.  4. 

36.  The  Equation.  If  two  expressions  represent  the  same 
number,  their  equality  may  be  indicated  by  the  sign  =  ;  such 
a  statement  of  equality  is  called  an  equation. 

Thus,  10  +  3  =  7,  2to  =  8,  j/  =  3  +  4,  and  2 a;  -f- 1  =  9  are  equations. 

37.  Members  of  an  Equation.  The  two  expressions  con- 
nected by  the  sign  of  equality  are  called  the  members  of  the 
equation. 

Thus,  in  wj  +  3  =  7,  to  +  3  and  7  are  members ;  they  are  called  re- 
spectively the  left  member  and  the  right  member,  or,  also,  the  first  mem- 
ber and  the  second  member. 

21 


22  A  HIGH   SCHOOL   ALGEBiv.. 

38.  Identical  Equation.  An  equation  that  either  involves 
no  letters,  or  that  is  true  for  any  values  whatever  that  may  be 
given  to  the  letters  involved,  is  called  an  identical  equation  or, 
briefly,  an  identity. 

Thus,  S  -\-4  =  1,  a  +  b  =  b  +  a,  5x  =  Sx  —  Sx,  SiTe  identities 

39.  Conditional  Equation.  An  equation  that  is  true  only  on 
condition  that  the  letter  or  letters  involved  have  particular 
values  is  called  a  conditional  equation  or,  briefly,  an  equation. 

Thus,  10  +  3  =  7  is  an  equation  that  is  true  only  on  condition  that  w 
represents  4. 

The  equation  is  an  abbreviated  sentence.  The  identical  equation  is  a 
declarative  sentence  and  states  that  the  two  members  are  necessarily  the 
same  —  differing,  at  most,  in  form.  The  conditional  equation  is  an  in- 
terrogative sentence,  and  asks  what  numbers  must  be  used  in  place  of 
the  letters  in  order  that  the  two  members  may  be  equal. 

ORAL  EXERCISES 

1.  What  must  be  added  to  3  to  make  10  ?     State  the  num- 
ber for  which  the  question  mark  stands  in  3  +  ?  =  10. 

2.  State  the  number  denoted  by  the  question  mark  in  each 
case:  5  +  ?  =  12;  ?+15  =  25;  60  =  45  +  ? 

3.  State  the  number  for  which  x  stands  in  each  of  the  fol- 
lowing: 5  +  a;  =  12;  a; -f  15  =  25;  60  =  45  +  ic;  25  + a;  =  40. 

State  the  number  denoted  by  the  question  mark  in  each 
case: 

4.  2  times  ?  =  12.  6.   i  of  ?  =  8.  8.   2  times  ?  +  1  =  9. 

5.  5  times?  =  30.  7.   fof?  =  12.  9.    3  times  ?  + 5  =  11. 

10.  State  the  number  represented  by  t  in  each  case :  2^  =  8; 
4^  =  32;  1^  =  10;  f  ^  =  30;  2^  +  1  =  11. 

11.  A  certain  number  less  4  is  20.     What  is  the  number  ? 

12.  A  certain  number  less  5  is  15.     ^hat  is  the  number  ? 

13.  a;  less  4  is  20.    What  is  a;?    2 a;  less  5  is  25.    What  is  2  a;? 

14.  In  3  a;  + 5  =  17,  what  is  3  a;?     What  is  a;? 
^        15.   In  4;)  + 2  =  10,  what  is  4p?     What  is  p? 


/ 


THE   EQUATION  23 

40.  In  equations  we  need  to  express  relations  between  num- 
bers by  algebraic  symbols. 

For  example,  a  number  that  is  greater  than  w  by  4  may  be  written  n  +  i. 

Or,  if  the  sum  of  two  numbers  is  11  and  one  of  them  is  n,  the  other  is 
expressed  by  11  — n. 

Or,  if  the  difference  of  two  numbers  is  15  and  the  larger  of  them  is  x, 
the  other  is  expressed  by  x  —  15. 

WRITTEN   EXERCISEIS 

Kepresent  a  number : 

1.  Greater  than  ?i  by  7.  7.   3  times  n  less  one. 

2.  Greater  than  n  by  3.  8.   3  greater  than  a  —b. 

3.  Less  than  n  by  5.  9.   a  greater  than  6  -j-  c. 

4.  Less  than  n  by  a.  10.   a  less  than  2  c  —  b. 

5.  Greater  than  2  n  by  6.       11.   6  greater  than  4  times  x. 

6.  2  b  less  than  5  times  n.     12.   2  a  —  1  greater  than  6  +  2. 
Write  the  other  part  if : 

13.  One  part  of  12  is  7.  16.    One  part  of  n  is  a. 

14.  One  part  of  x  is  3.  17.   One  part  of  a  is  n. 

15.  One  part  of  10  is  y.  18.   One  part  oi  a  —  b  is  x. 

19.  The  diiference  between  two  numbers  is  8.  The  smaller 
of  the  two  is  12.     Write  the  larger. 

20.  The  difference  between  two  numbers  is  d.  The  smaller 
of  them  is  a.     Write  the  larger. 

21.  The  difference  between  two  numbers  is  10.  .The  larger 
is  25.     Write  the  smaller. 

22.  The  difference  between  two  numbers  is  21.  The  larger 
is  y.     Write  the  smaller. 

23.  The  difference  between  two  numbers  is  d.  The  larger 
is  2  c.     Write  the  smaller. 

24.  4  n  plus  a  number  is  26.     Write  the  number  in  symbols. 

25.  The  sum  of  three  numbers  is  45.  One  of  them  is  6, 
another  is  a.     Write  the  third. 

26.  The  sum  of  three  numbers  is  s.  One  of  them  is  a, 
another  2  b.     Write  the  third. 

3 


24  A  HIGH  SCHOOL  ALGEBRA 

Express  by  how  much : 

27.  18  exceeds  a.  31.   12  is  less  than  n. 

28.  17  exceeds  2  c.  32.   15  is  less  than  2  a. 

29.  24  exceeds  a  —  h.  33.   44  is  less  than  h  +c. 

30.  36  exceeds  2  a  — c,  34.   35  +  a  exceeds  6  +  c. 
If  A  is  n  years  old,  express  his  age : 

35.  5  years  ago.  38.   x  years  hence. 

36.  X  years  ago.  39.    a  —  3  years  ago. 

37.  5  years  hence.  40.    2  a  — 5  years  hence. 

41.  The  translation  of  algebraic  expressions  into  words 
assists  in  interpreting  equations. 

EXAMPLES 

1.  w  —  3  =  7  is  translated  "a  number  less  three  is  seven." 

2.  2  X  —  1  =  5  is  translated  "  twice  a  number  less  one  is  five." 

3.  3a;  — 2  =  5aj  — Sis  translated  "three  times  a  number  less  two 
equals  five  times  the  number  less  eight." 

4.  2  w  —  m  =  4  is  translated  "twice  a  number  less  a  second  number 
is  four." 

b.  A-=\  ah,  referring  to  a  triangle,  is  translated  "the  area  of  a  tri- 
angle is  one  half  the  product  of  the  base  and  altitude." 

6.  F=  Trr'-^a,  referring  to  a  circular  cylinder,  is  translated  "  the  volume 
of  a  cylinder  is  pi  times  the  square  of  the  radius  times  the  altitude." 

ORAL   EXERCISES 

Translate  into  words : 

1.  n-f5  =  8.  8.    n,  n  +  1,  w  +  2. 

2.  a;-l  =  10.  9.   2  w,  2n  + 2,  2  n  +  4. 

3.  3  a; -2  =  12.  10.    n,n  —  l,n-2. 

4.  5a;-l  =  a;  +  4.  11.    2  n,  2  71- 1,  2  n- 3. 

5.  m  +  n  =  17.  12.   2a;-l  =  26H-l. 

6.  m-n  =  l.  13.   3i9-f6  =  4g-6. 

7.  2a;  +  32/  =  15.  14.  pq-\-1^2pq-10. 

15.  Taking  A  to  mean  the  area  of  a  rectangle,  read  A  =  ah. 

16.  Taking  V  to  mean  the  volume  of  a  rectangular  solid, 
of  dimensions  a,  h,  and  c,  read  F=  abc. 


THE   EQUATION  25 

17.  Taking  A  to  mean  the  area  of  a  circle,  read  A  =  irr^. 

18.  Taking  C  to  mean  the  circumference  of  a  circle,  read 
C=2nr, 

19.  Taking  I  to  mean  simple  interest,  read  I=prt.     Also 

read  t  =  —  ' 
pr 

42.  Substitution.  A  number  symbol  put  in  place  of  another 
is  said  to  be  substituted  for  it. 

For  example  : 

6  a  +  2  becomes  5-3  +  2  when  3  is  substituted  for  a ;  and  aa;  +  7  be- 
comes a?)  +  7  when  h  is  substituted  for  x. 

43.  Unknown.  A  number  symbol  whose  value  is  not 
known  is  called  an  unknown  number,  or  simply  an  unknown. 

44.  Satisfying  an  Equation.  If  an  equation  becomes  an 
identity  when  certain  numbers  are  substituted  for  the  un- 
knowns, the  numbers  substituted  are  said  to  satisfy  the 
equation. 

Thus,  5  is  said  to  satisfy  the  equation  3  x  =  15,  because  3  x  5  =  15. 
The  equation  is  not  satisfied  by  any  other  number,  because  3  times  any 
other  number  is  not  15.  Also,  7  and  5  are  said  to  satisfy  the  equation 
Sx  +  2y  =  Sl,  because  3  •  7  +  2  •  5  =  31. 

45.  Root  of  an  Equation.  A  number  that  satisfies  an  equa- 
tion is  called  a  root  of  the  equation. 

46.  Solving  Equations.  To  solve  equations  is  to  find  their 
roots. 

ORAL  EXERCISES 

What  number  satisfies  each  of  the  following  equations  ? 

1.  Sx  =  6.  7.    7  2/4-5  =  40.  13.  2  7i  =  90. 

2.  9a;  =  18.  8.   2.^  +  1=3.  14.  22/  +  7  =  13. 

3.  7cc  =  35.  9.    30-6  =  4  2/.  15.  J  w  +  3  =  2. 

4.  4  a;  =  32.  10.    4w;  +  2  =  10.  16.  2?i  =  4800. 

5.  5a;  +  2  =  22.         11.    4  10  +  6  =  46.  17.  2  7^4-1  =  27. 

6.  8aj  +  12=20.        12.   ^z-\-3  =  d.  18.  2  7i  +  1  =  625. 


26  A  HIGH  SCHOOL  ALGEBRA 

Solve  the  equations : 

19.  4a;  =  20.  24.  6u  =  U-2.  29.  9r  =  360. 

20.  32/  +  4:  =  25.         25.  8  +  2  =  5s.  30.  17%  =  3400. 

21.  4^  +  1  =  27.         26.  7a;  +  7  =  28.  31.  20^  =  50-10. 

22.  2r4-5  =  13.         27.  14  =  6a;  +  2.  32.  60  +  lo=25x, 

23.  4'v  +  l  =  9.  28.  20-4  =  4?/.  33.  182/  =  360. 

47.  Preparatory. 

If  two  weights  are  in  balance,  and  if  the  following  changes 
are  made  in  one  weight,  what  change,  in  each  case,  must  be 
made  in  the  other  to  preserve  the  balance  ? 

1.   Two  ounces  added.  2.   Two  ounces  taken  away. 

3.  The  number  of  ounces  in  one  weight  made  three  times 
as  great. 

4.  The  number  of  ounces  in  one  weight  made  ^  as  great. 

48.  Properties  used  in  Solving  Equations.  The  preceding 
exercises  suggest  the  following  properties  : 

1.  If  the  same  number  is  added  to  equal  numbers,  the  results 
are  equal. 

2.  If  the  same  number  is  subtracted  from  equal  numbers,  the 
results  are  equal. 

3.  If  equal  numbers  are  multiplied  by  the  same  number,  the 
res^dts  are  equal. 

4.  If  equal  numbers  are  divided  by  the  same  number  (not  zero), 
the  results  are  equal. 

Note.  The  reason  for  excluding  zero  as  a  divisor  is  explained  in 
Chapter  IX. 

49.  The  following  examples  show  how  these  properties  are 
used  in  solving  equations  : 

EXAMPLES 
1.    Solve:  3x  +  5  =  23.  (1) 

Subtracting  5  from  both  members,  Bx  =  18.  (^) 

Dividing  both  members  of  {3)  by  3,  X  =  6.  (5) 

Test.        6  satisfies  3  a;  +  5  =  23,  because  3  •  6  +  5  =  23. 


THE  EQUATION"  27 

2.    Solve:  ^4-2 +^p=  2p_j_6.         (^j) 

Subtracting  §/»  from  both  members,  |j3  +  2  =  6.  (2) 

Subtracting  2  from  both  members  of  (2),  f  p  =  4.  (5) 

Dividing  both  members  of  (3)  by  §,  p  =  6.  (-^) 

Test.  6  satisfies  {!) ,  because  6  +  2  +  ^  •  6  =  f  •  6  +  6. 

10  =  10. 

50.  Testing.  The  correctness  of  the  work  of  solving  an 
equation  should  be  tested  by  substituting  the  result  in  the 
given  equation.  If  the  members  become  identical,  the  number 
substituted  is  a  root  of  the  equation. 

WRITTEN   EXERCISES 

Solve  and  test : 

1.  4a;  +  l  =  7.  5.  16^  +  5  =  37.  9.   3.«  +  2=19. 

2.  3aj  +  l  =  10.  6.  14a;  =  25  4-9a;.       10.   2  +  42  =  85. 

3.  5a;  =  x  +  16.  7.  48  =  8t/  +  16.  11.   15s+2  =  12. 

4.  2a;  +  7  =  27.  8.  12^  +  13  =  49.         12.   ia;-f2=f 

13.  62/  +  2=20.  20.    11m  +  3  =  2m +  9  +  2m. 

14.  82  +  2  =  42.  21.   8j9  +  5  =  2p  +  14  +  3i). 

15.  72  =  12ic.  22.   122  +  28  =  72  +  53. 

16.  6a:  =  9  +  3a;.  23.   40  +  32  =  58  +  2. 

17.  fa;  =  25  +  ia;.  24.    |2/  +  45  =  |2/  +  55. 

18.  ll2/  +  l  =  92/  +  3.  25.    4  ^,  +  31  =  f  ^^ +  41. 

19.  3a;+2ic  =  4a;  +  16.  26.   ^aj  +  20  =  32 +  f a;. 

51.  Use  of  the  Equation  in  Solving  Problems.  Equations 
may  be  used  in  solving  problems. 

EXAMPLES 

1.   If  a  certain  number  is  doubled  and  16  is  added  to  the 
product,  the  result  is  46.     What  is  the  number  ? 
Solution.     1.   Let  n  be  the  number. 

2.  Then  2  n  +  16  is  double  the  number  plus  16. 

3.  But  46  is  given  as  double  the  number  plus  16. 

4.  Therefore,  2  n  +  16  =  46. 

5.  Therefore,  2  w  =  30,  and  n  =  15.     Why  ? 

Test.     15  doubled  makes  30,  and  30  plus  16  is  46. 


28  A   HIGH   SCHOOL   ALGEBRA 

2.  A  salesman  sold  twice  as  many  articles  on  Friday  as  on 
Thursday,  and  5  more  on  Saturday  than  on  Friday ;  on  Satur- 
day he  sold  15.     How  many  did  he  sell  on  Thursday  ? 

Solution.     1.    Let  x  be  the  number  that  he  sold  on  Thursday. 

2.  What  does  2  x  represent  ?     2  ic  +  5  ? 

3.  State   two   expressions,  each  of  which  is  the  number 

sold  on  Saturday. 

4.  Since  2  x  +  5  =  15,  x  =  5, 
Test.     2  •  5  +  5  =  15. 

Any  letter  may  be  used  to  represent  the  unknown,  as  y  for  the  num- 
ber of  years,  d  for  the  number  of  dollars,  r  for  rate,  or  p  for  the  number 
of  pounds  pressure  as  in  physics,  but  in  algebra  x  is  most  frequently  used. 

52.  Finding  the  Equation.  In  each  solution  above,  step  4 
contains  the  statement  of  the  problem  in  the  form  of  an  equa- 
tion. This  statement  is  reached  by  finding  two  expressions  for 
the  same  number  and  using  them  as  the  members  of  an  equation. 

53.  Sign  of  Deduction.  Instead  of  the  word  "hence,"  or 
<^  therefore,"  the  sign  .'.is  often  used.  It  is  called  the  sign  of 
deduction. 

Thus,  .  •.  2  w  -1- 16  =  46  is  read,  "  Therefore,  2  w  -1-  16  =  46." 

WRITTEN   EXERCISES 

Write  the  solution  of  each  problem  in  steps  as  shown  above : 

1.  A  house  and  lot  are  worth  $  4800,  and  the  house  is 
worth  7  times  as  much  as  the  lot.     Find  the  value  of  -each. 

2.  Lucy  thought  of  a  number,  doubled  it,  added  16,  and 
obtained  50.     Of  what  number  did  she  think  ? 

3.  The  continued  height  of  a  tower  and  flagstaff  is  120  ft. ; 
the  height  of  the  tower  is  5  times  that  of  the  flagstaff.  Find 
the  height  of  each. 

4.  I  of  the  total  height  of  a  bridge  pier  is  out  of  the  water, 
and  10  ft.  of  the  height  is  under  water.  AVhat  is  the  height 
of  the  pier  ? 

5.  When  goods  are  sold  at  a  gain  of  ^  of  their  cost,  what 
is  the  cost  of  goods  which  sell  for  $  12  ? 


THE   EQUATION  29 

6.  A  man's  salary  was  increased  by  J  of  itself;  he  then 
received  $  1600.     What  was  his  salary  before  the  increase  ? 

7.  A  merchant  gained  in  one  year  an  amount  equal  to  J  of 
his  capital.  He  then  had  $  6250.  How  many  dollars  had  he 
at  the  beginning  of  the  year  ? 

8.  The  area  of  Kansas  is  twice  that  of  Ohio.  The  sum  of 
their  areas  is  123,000  sq.  mi.     Find  the  area  of  each. 

9.  A  freight  train  consisted  of  48  cars.  The  number  of 
closed  cars  was  6  more  than  twice  the  number  of  open  cars. 
Find  how  many  there  were  of  each. 

REVIEW 
WRITTEN   EXERCISES 

1.  Jg-  of  the  distance  from  Boston  to  Cincinnati  is  441  mi. 
Find  the  distance  between  these  two  cities. 

2.  The  product  of  a  certain  number  and  13  is  221.  Write 
an  equation  expressing  this  fact,  and  find  the  number. 

3.  3  times  a  certain  number  increased  by  5  equals  twice  the 
number  increased  by  17.     Find  the  number. 

4.  3  times  a  number  plus  3  equals  J  of  the  number  plus  27. 
Find  the  number. 

5.  In  a  recent  year  France  mined  twice  as  much  coal  as 
Russia,  and  together  they  produced  48  million  tons.  Plow 
many  tons  did  each  produce  ? 

6.  In  a  recent  year  the  United  States  mined  12  times  as 
much  coal  as  Belgium,  and  together  they  produced  286  million 
tons.     How  many  tons  did  each  produce  ? 

7.  The  length  of  a  garden  was  3  times  its  width,  and  the 
distance  around  it  was  72  yd.     Find  its  length  and  width. 

8.  If  a  tennis  ball  rebounds  f  of  the  height  from  which  it 
was  dropped,  from  what  height  must  it  be  dropped  to  rebound 
3i  ft.  ? 


30  A   HIGH   SCHOOL   ALGEBRA 

SUMMARY 

The  following  questions  summarize  the  definitions  and  pro- 
cesses treated  in  this  chapter  : 

1.  Define  an  equation.  Sec.  36. 

2.  What  are  the  members  of  an  equation  ?  Sec.  37. 

3.  What  is  an  identical  equation  9  Sec.  38. 

4.  What  is  a  conditional  equation  9  Sec.  39. 

5.  What  is  meant  by  substitution  .?  Sec.  42. 

6.  What  is  an  unknown  .^  Sec.  43. 

7.  Explain  the  meaning  of  satisfy  an  equation.  Sec.  44. 

8.  Define  a  root  of  an  equation.  Sec.  45. 

9.  Define  to  solve  an  equation.  Sec.  46. 

10.  What  are  the  properties  used  in  solving  equations  ? 

Sec.  48. 

11.  How  is  the  correctness  of  the  work  in  solving  equations 
tested  f  Sec.  50. 

12.  How  is  the  solution  of  a  problem  tested  ?  Sec.  51. 

13.  How  is  a  problem  stated  in  the  form  of  an  equation  ? 

Sec.  52. 

14.  What  is  the  sign  of  deduction  ?  Sec.  53. 


CHAPTER   IV 
RELATIVE  NUMBERS 

54.  Numbers  used  in  Algebra.  The  preceding  work  is 
much  like  that  of  arithmetic ;  in  fact,  it  might  be  called  literal 
arithmetic.  We  take  up  now  a  class  of  numbers  belonging  to 
algebra  proper,  and  the  following  examples  will  illustrate  them : 

I.   Distances  measured  in  opposite  directions. 

1.  40  ft.  up  an  elevator  shaft +  20  ft.  down  the  shaft  is  the 
same  as  how  many  feet  up  the  shaft  ? 

2.  20  ft.  up +40  ft.  down  is  the  same  as  how  many  feet  down  ? 
Similarly,  to  what  is  each  of  the  following  equivalent  ?    • 

3.  40  rd.  traveled  to  the  right  +  20  rd.  traveled  to  the  left. 

4.  20  rd.  traveled  to  the  right  +  20  rd.  traveled  to  the  left. 

5.  20  rd.  traveled  to  tlie  right  +  50  rd.  traveled  to  the  left. 

II.  Rise  and  fall  of  temperature. 

1.  Calling  rise  of  temperature  R  and  fall  of  temperature  F 
15°  R-^10°F  is  the  same  as  (?)°i2. 

Similarly : 

2.  10°i2  +  15°i^=?  4.   30°i^+15°i2  =  ? 

3.  35°i<^+45°i2  =  ?  5.   40°  i2  +  40°i?^=  ? 

III.  Amounts  gained  and  lost. 

1.    $22  gain  +  $26  loss  =  ?  loss. 
Similarly,  using  G  for  gain  and  L  for  loss : 

2.  $40(^+$30i  =  ?  4.    $35Z+S20G^  =  ? 

3.  $17 (^+$171,  =  ?  5.    $456?+$15  7v=? 

55.  Relative  Numbers.  In  each  of  the  preceding  illustra- 
tions we  have  considered  quantities  which  had  two  opposite 
directions,  or  senses.  Numbers  which  measure  quantities  hav- 
ing opposite  senses  are  called  relative  numbers. 

31 


32  A   HIGH   SCHOOL   ALGEBRA 

56.  In  algebra  we  distinguish  two  opposite  senses  by  calling 
one  positive  and  the  other  negative.  Either  may  be  called  posi- 
tive, but  the  opposite  to  the  positive  is  always  called  negative. 

For  example : 

If  distance  upward  is  called  positive,  distance  downward  is  called  nega- 
tive.    If  a  rise  of  temperature  is  called  positive,  a  fall  is  called  negative. 

57.  Positive  and  Negative  Number.  A  number  that  meas- 
ures a  quantity  taken  in  the  positive  sense  is  called  a  positive 
number ;  one  that  measures  a  quantity  taken  in  the  negative 
sense  is  called  a  negative  number. 

ORAL  EXERCISES 

What  must  be  taken  as  negative  when  each  of  the  following 
is  taken  as  positive  ? 
^Number  of :  ' 

1.  Feet  to  the  right.     3.    Dollars  gained.     5.  Points  won. 

2.  Miles  southward.     4.    Degrees  upward.   6.  Pounds  lifted. 

58.  Notation  for  Positive  and  Negative  Numbers.  Since  a 
number  added  is  offset  by  the  same  number  subtracted,  and  rel- 
ative numbers  similarly  offset  each  other,  the  signs  +  and  — 
are  used  to  designate  positive  and  negative  numbers  respectively. 

Thus: 

+  3  means  3  positive  units,  and  denotes  3  units  to  be  added. 

—  3  means  3  negative  units,  and  denotes  3  units  to  be  subtracted. 

59.  Signs  of  Character  and  Signs  of  Operation.  Thus,  in 
algebra  the  signs  -f-,  — ,  are  used  to  indicate  the  operations  of 
adding  or  subtracting  numbers,  and  also  to  indicate  the  positive 
or  negati ve'c/iarac^er  of  numbers. 

If  it  is  necessary  to  distinguish  a  sign  of  character  from  a  sign 
of  ojjeration,  the  former  is  put  into  a  parenthesis  with  the  number 
it  affects.  ^.       ^ 

Thus,  +  8  —  (—3^  mea^is :  positive  8  minus  negative  3- 

When  no  sign  of  character  is  expressed,  the  sign  plus  is  understood. 

Thus,  5  —  3  means  :  positive  5  minus  positive  3. 
Similarly,  8  a  +  9  a  means  :  positive  8  a  plus  positive  9  a. 


9 


RELATIVE  NUMBERS  33 

60.  Signed  Numbers.  In  algebra  every  number  is  under- 
stood to  have  either  the  sign  -f  or  the  sign  — .  Consequently 
the  numbers  of  algebra  are  often  called  signed  numbers. 

61.  Absolute  Value.  The  value  of  a  signed  number  apart 
from  its  sign  is  called  its  absolute,  or  numerical,  value. 

Thus,  6^  is  the  absolute  value  of  6°  above  zero  or  6°  below  zero. 
And  6  is  the  absolute  value  of  either  +  6  or  —  6. 

ORAL  EXERCISES 

Read  the  following  in  full,  in  accordance  with  Sec.  59 : 

1.   7-4.  6.   14 -(-6).  11.  6-8. 

2.-6-8.       -  7.    -14^+6)".'']^,  12.  6  +  8. 

3.    -8  +  25.  8.    -6*-(-8).   li,  13.  -6 +  (-8). 

4.-2  +  7.  9.    7-15.  ^  14.  -4 +(-9). 

5.   9-(-3).  10.    -7-(+2).  15.  15-(-9). 

WRITTEN   EXERCISES 

Indicate,  using  the  signs  +,  —  : 

1.  The  sum  of  positive  6  and  positive  4. 

2.  The  sum  of  positive  a  and  negative  b. 

3.  The  difference  of  positive  x  and  positive  y. 

4.  The  difference  of  negative  6  and  positive  3. 

5.  The  difference  of  negative  a  and  positive  b. 

6.  The  sum  of  negative  c  and  negative  d. 

62.  Preparatory. 

1.  4  points  won  +  3  points  won  = points  won. 

2.  4  points  lost  +  5  points  lost  = points  lost. 

3.  7°  above  zero  +  9°  above  zero  = degrees  above  zero. 

63.  Addition  of  Numbers  having  Like  Signs.  When  the 
numbers  added  are  positive,  the  sum  is  positive ;  and  when 
they  are  negative,  the  sum  is  negative. 

TJierefore,  to  add  either  positive  numbers  or  negative  numbers, 
find  the  sum  of  their  absolute  values  and  prefix  the  corresponding 
sign.  ^_^ 


34  A   HIGH  SCHOOL  ALGEBRA 

ORAL  EXERCISES 


Add: 

1.   +2 

6. 

+  6 

9.    +2  a 

13. 

-6x 

+  3 

+  5 

+  3a 

-5x 

2.    -2 

6. 

-6 

10.    -2a 

14. 

-5y 

-3 

-5 

-3a 

(    )  « 

-Sy 

3.    +4 

7. 

-4 

11.    +46 

15. 

-10  a; 

+  3 

-7 

+  36 
(    )& 

-    5a; 

4.-4 

8. 

-5 

12.    -46 

16. 

-12nx 

-3 

-8 

-36 

-   Snx 

64.   Number  Pictures  or  Graphs.     Numbers  are  often  repre- 
sented by  lines.     Such  representations  are  called  graphs. 


Thus,  the  line  AB  represents  5,  the  line  a  represents  4,  and  the  line  b 
represents  6. 

65.    Graphical  Addition.     Positive  and  negative  numbers  may 
be  arranged  on  a  straight  line  as  follows : 


—4      -3     -2      -1         0       -f1       +2      +3     +4      +5 

POSJTtVE  SENSE 

This  arrangement  is  called  the  number  scale,  and  it  may  be 
used  to  perform  additions  graphically. 

EXAMPLES 

1.   To  add  +  2  and  +  3  :  Let  a  moving  point  start  at  0  and  proceed  2 

2^3  units  in  the  positive  direction  (to  the 

!        ,        .        ,        ,      ^i       right),  and  from  the  place  where  it 


~2— 1  0  1  2  3  4  5  then  is,  proceed  3  units  farther  in 
the  positive  direction.  The  final  position  of  the  moving  point  will  be 
distant  2  +  3  units  from  the  starting  point.     That  is,  2  +  3  =  5. 


RELATIVE   NUMBERS  35 

2.   To  add  —  2  and  +  5  :  Let  the  point  proceed  2  units  in  the  negative 
sense   (toward  the   left),  and  from   there  5 


units  in  the  positive  sense.     The  final  posi-      f<      "^    \ 


-5 

.     r 

-1-2      ^ 

-3 

—  2 

-1        0 

1         : 

>         3 

tion  is  three  positive  units  from  the  starting  -2.  -l     o     12     3     4     5 
point.     That  is,  -  2  +  5  =  3.  -2+5 

3.   To  add  +•  2  and  —  6  :  Proceed  2  units  to  the  right,  and  from  there  5 
units  to  the  left.     The  final  position  of 
the  moving  point  is  three  units  to  the 
left  of  the  starting  point.    That  is, 
2+(-5)  =  -3. 

2  +  -^  4.   Similarly,  to  add  (-  2)  and  (-  3) 

proceed  two  units  to  the  left,  and  from  there  three  units  to  the  left. 
The  final  position  is  5  units  to  the  ,g  -2      1 

left.  i*      ■       .       I*     "■      "I       . 

Thatis,  (-2)  and  (-8)  =  -5.  '^     "^    "^     -2-10       1 


WRITTEN   EXERCISES        / 

Add  by  means  of  the  number  scale  as  above : 

1.  3  +  4.  4.    -2  +  8.      7.    4+(-3).     10.    -2+(-5). 

2.  8  +  3.  5.    -2  +  9.      8.    2  +  (-6).     11.    -3+(-3). 

3.  -2  +  5.      6.    -9  +  5.      9.   7+(-3).     12.    -4+(-7). 

66.   Preparatory. 
1.    Add  5  and  —  3. 

Regard  5  as  made  up  of  +  3  and  -f  2  ;  then 

+3+2]  r      5 

■3 


0+2 J  [       2 

That  is,  the  -  3  offsets  +  3  of  the  +  5,  and  the  sum  is  +  2. 

2.   Add  -  8  and  6.         ^  ^  -^ 

Regard  —  8  as  made  up  of  —  6  and  —  2  ;  then 
-6-2 

+  6 
0-2 

That  is,  the  +  6  offsets  —  6  of  the  —  8,  and  the  sum  is  —  2. 
Zero,  as  in  arithmetic,  means  the  difference  between  two  equal  num- 
bers like  3  —  3  ;  hence,  in  accordance  with  Sec.  58,  3  +  (—  3)  =  0. 
Thus  the  addition  of  zero  has  no  effect ;  6+0  =  6,  or— 6+0=—  6. 


36  A   HIGH   SCHOOL   ALGEBRA 

67.  Addition  of  Numbers  with  Unlike  Signs.  In  adding  a 
positive  and  a  negative  number,  a  positive  unit  and  a  negative 
unit  offset  each  other. 

Therefore,  to  add  a  positive  and  a  negative  number  find  the 
difference  of  their  absolute  values  and  prefix  to  it  the  sign  of  the 
number  having  the  greater  absolute  value. 

ORAL  EXERCISES 

State  the  suras : 

1.  7  negative  units  +  4  positive  units. 

2.  7  negative  units  +  12  positive  units. 

3.  8  negative  units  +  7  positive  units. 

4.  9  positive  a's  +  9  negative  a's. 

5.  10  positive  a;'s  +  15  negative  aj's. 

WRITTEN   EXERCISES 

Add,  separating  the  numbers  as  in  Sec.  QQ: 


1. 

5 

5. 

-3 

9. 

-9 

13.    -40 

-2 

7 

3 

30 

2. 

6 

6. 

-5 

10. 

-6 

14.    -25 

-4 

9 

2 

17 

3. 

8 

7. 

-6 

11. 

-12 

15.        43 

-5 

12 

8 

-65 

4. 

10 

8. 

-11 

12. 

-17 

16.        39 

-7 

20 

9 

-44 

17. 

-9a 

20. 

-Ibp 

23 

.    -  12  a6 

4-5a 

+  10p 

+  10a6 

18. 

+  96 

21. 

-?,x 

24 

.     —  8  mn 

-56 

+  9a; 

+  ^mn 

19. 

+  96 
-96 

22. 

+    8a; 
-ISx 

25 

.       +4a!?/ 
-Sxy 

RELATIVE  NUMBERS 


37 


18.   Preparatory. 

1.  5+?  =  8. 

2.  5  +  ?  =  3. 


5  +  ?  =  3. 


6.  Question  1  may  be  read  "S  less  5  are  how  many?  " 
Read  the  questions  in  2,  3,  and  4,  and  state  the  difference  in 
each  case. 

69.   Subtraction  of  Positive  and  Negative  Numbers.     The 

difference  is  the  number  which  added  to  the  subtrahend  pro- 
duces the  minuend. 

Thus :  6  less  -  4  =  10    because     -  4  +  10  =  6. 

—  15  less  —  7  =—  8  because     —  7  plus  --  8  =—  15. 
4  a  less  —2  a  =  Q  a  because    — 2a4-6a  =  4a. 

The  terms  subtrahend  and  minuend  are  used  as  in  arithmetic,  the 
former  to  mean  the  number  taken  away  and  the  latter  the  number  from 
which  the  subtrahend  is  taken. 


(.\1) 


ORAL  EXERCISES 

State  the  numbers  to  fill  the  blanks : 


2. 


3. 


4. 


5. 


15-8  =  (). 
8-()  =  2, 
2-8  =  ()- 
12-()  =  -l, 
.-1-12  =  (). 

f-7  +  ()  =  3, 
l3-(-7)  =  (). 

-6  +  ()  =  8, 
8-14  =  0. 


f-5  +  ()=20, 
■   l20-25  =  (). 

f-6  +  () 9, 

•    |_9-(-3)  =  (). 


8. 


U  +  ()  =  -15, 
15-(-4)  =  (). 


|_r-(-21)  =  (). 

.0.  {1^+Q^-^' 


5-(-20)  =  (5. 


11.  20-8  =  (). 

12.  8-20=(). 

13.  3-(-9)  =  (). 


14.  16-(-5)  =  (). 

15.  _17-13  =  (). 

16.  _16-(-5)  =  (). 


38  A   HIGH   SCHOOL   ALGEBRA 

70.  We  have  shown  in  Sec.  58  that  a  negative  and  a  posi- 
tive unit  offset  each  other.  Hence,  to  subtract  1  is  the  same 
as  to  add  —  1,  and  vice  versa;  and  to  subtract  any  number  a 
is  to  add  —  a,  and  vice  versa. 

Thus  :  7  less  5  =  2  is  the  same  as  7  plus  —  5  =  2, 

or,  7  less  —  5  =  12  is  the  same  as  7  plus  5  =  12. 

By  its  meaning  the  subtraction  of  zero  has  no  effect.     (Sec.  66.) 

71.  To  subtract  a  number,  change  its  sign  and  add  it. 

ORAL  EXERCISES 

Find  these  differences  by  adding : 
1.      6  5.     12  9.      7a  13.    -4a 

-3  -9  4a  -7a 


2. 


(  )« 

(    )« 

3 

6. 

15 

10.      4  a 

14. 

8a; 

-8 

-15 

7a 
(   )« 

Ix 

(  )« 

5 

7. 

■7 

11.       7  a 

15. 

1  X 

-3 

-15 

-4a 

%x 

10 

8. 

8 

12.  —4a 

16. 

-Ix 

-5 

-17 

7a 

-8a? 

3. 


Perform  the  indicated  operations  : 

17.  13 -(-5) +  8.  19.    -15 +  (-8)- (4- 22). 

18.  13-(-6)  +  (-4).  20.    _17-9-(-20). 

72.   Preparatory. 

1.  A  man  earned  $3  on  Monday  and  $3  on  Tuesday.     How 
many  dollars  did  he  earn  in  the  two  days  ? 

2.  To  multiply  3  by  2  is  to  take  3  how  many  times  as  the 
addend  ? 

3.  A  man  lost  $  3  on  Monday  and  $  3  on  Tuesday.     How 
many  dollars  did  he  lose  in  the  two  days  ? 

4.  To  multiply  —  3  by  2  is  to  take  —  3  how  many  times  as 
an  addend  ? 


RELATIVE  NUMBERS  39 

73.  Mixltiplication    of    Positive    and    Negative    Numbers. 

Multiplication  by  a  positive  integer  means  taking  the  multiplicand 
as  an  addend  as  many  times  as  there  are  units  in  the  midtiplier. 

Correspondingly,  multiplication  by  a  negative  integer  means 
taking  the  multiplicand  as  a  subtrahend  as  many  times  as  there 
are  units  in  the  multiplier. 

For  example : 
4  multiplied  by  3  =  4  +  4  +  4  =  12. 

-  4  multiplied  by  3  =-  4  +  (-  4)  +  (-  4)  =  -  12. 
4  multiplied  by-3=-4-4-4=-12. 

-  4  multiplied  by  -  3  =-  (-  4)  -  (-  4)  -  (-  4)  =  +  4  +  4  +  4  =  12. 

The  numerical  value  of  the  product  is  the  product  of  the  nu- 
merical values  of  the  factors. 

74.  The  law  of  signs  in  multiplication,  which  applies  to  inte- 
gral and  fractional  numbers  alike,  may  be  stated  thus :  If  both 
factors  are  positive  or  if  both  are  negative,  their  product  is 
positive.  If  one  is  positive  and  the  other  negative,  their  prod- 
uct is  negative. 

In  symbols  :        +  times  +  =  +  —  times  +  =  — . 

—  times  —  =  +  +  times  —  =  — . 

75.  This  law  is  easily  remembered  in  the  form : 

Tlie  product  of  two  factors  of  like  signs  is  positive,  and  of  two 
factors  of  unlike  signs  is  negative. 


ORAL  EXERCISES 

State  the  product  in  each  of  the  following : 


1. 

+  5. +3. 

7. 

-7.-6. 

13. 

-8.+i. 

2. 

-5  .  +3. 

8. 

-8.-8. 

14. 

+  3.  -4. +5. 

3. 

+  5.-3. 

9. 

-  7  .  -  11. 

15. 

_5.  +2  .  -3. 

4. 

-5.-3. 

10. 

-12. +6. 

16. 

+  14. +1 

5. 

-7.  +7. 

11. 

+  5  .-11. 

17. 

+  2.-3.-4. 

6. 

+  8.-9, 

12. 

+  6.-|. 

18. 

_4  .  -5.  -3. 

40 


A   HIGH  SCHOOL  ALGEBRA 


76.  Division  of  Positive  and  Negative  Numbers.  We  have 
seen  in  arithmetic  that 

Quotient  x  Divisor  =  Dividend. 

Thus,  we  know  that  24  -j-  8  is  3,  because  3  x  8  =  24. 

Applying  this  relation  to  division  with  positive  and  negative 
numbers,  we  can  determine  the  sign  of  the  quotient.  There 
are  four  cases,  as  follows  : 

(  ). 


24--  +  8  =  ();-24--8i 

=  (  ) ;  +  24  - 

--8=(  );  _24-f-  +  8 

According  to  Sec.  74  : 

+  24--  +  8=+3 

because 

+  3  X  +  8  =  +  24. 

_24--8  =  +  3 

because 

+  3  X  -  8  =  -  24. 

+  24--8=-3 

because 

_  3  X  -  8  =  +  24. 

_24--  +  8=-3 

because 

-  3  X  +  8  =  -  24. 

In  symbols : 

^=  +  ;  - 

=  +  ;  ^  = 

-  •   — =  — . 

+           - 

+ 

77.   This  law  is  easily  remembered  in  the  form : 
The  quotient  of  two  numbers  of  like  signs  is  positive,  and  the 
quotient  of  two  numbers  of  unlike  signs  is  negative. 


State  the  quotient : 

1.  _6-  +  3. 

2.  _12--  +  4. 


16 


3. 
4. 
5.    +15 


+  20 ---5. 


3. 


ORAL  EXERCISES 


6.  -8- -8. 

7.  -8 --  +  8. 


4. 


8. 

9. 

10. 


15 


_36^_6 


35 


-5. 


+  7. 


11.  +10-^.-4. 

12.  +18 ---12. 

13.  20 ---4. 

14.  30 --10. 

15.  -42 --  +  7. 


Perform  the  indicated  operations : 

16.  6(-3)-^9. 

17.  4(-6)-^-12. 

18.  18--(_3)^-2. 

19.  4(-9)-3x3-(18-^-2). 

20.  6  x  9-4x9 -(-24^-4). 


RELATIVE  NUMBERS  41 

78.  The  Greater  of  Two  Numbers.  Of  two  given  numbers 
that  one  is  the  greater  which  can  be  produced  by  adding  a 
positive  number  to  the  other.     The  other  number  is  the  less. 

For  example  : 

11  is  greater  than  8  because  it  is  necessary  to  add  +  3  to  8  to  make  11. 

7  is  greater  tlian  —2  because  it  is  necessary  to  add  +9  to  —2  to  make  7. 

—  4  is  greater  than  —  9  because  it  is  necessary  to  add  +  5  to  —  9  to 
make  —4. 

79.  The  symbol  >  is  read  "  is  greater  than,"  and  <  is  read 
"is  less  than." 

For  example  : 

8  >  2  is  read  "  8  is  greater  than  2." 

—  1  >  —  5  is  read  "  —  1  is  greater  than  —  5." 

—  5  <,  —  1  is  read  *'  —  5  is  less  than  —  1." 

The  terms  "algebraically  greater"  and  "algebraically  less"  are  used 
when  positive  and  negative  numbers  are  compared.  The  terms  "  numeri- 
cally greater"  and  "numerically  less"  apply  to  absolute  values. 

For  example  : 

4  is  greater  than  —  9,  but  4  is  numerically  less  than  —  9. 

—  6  is  greater  than  —  15,  but  —  15  is  numerically  greater  than  —  6. 

—  2  is  algebraically  greater  than  —  12,  but  numerically  less  than  —  12. 


ORAL  EXERCISES 

Read  the  following  and  state  why  each  is  correct  : 

1.  7  >  5.  3.    -  2  >  -  5.    5.    3  <  5.  7.    -  1  <  0. 

2.  4  >  -  8.     4.    0  >  -  7.  6.    -  4  <  2.     8.    -  8  <  -  6. 
From  the  following  list  select  the  numbers  that  are : 

a.  Greater  than  6.  d.   Numerically  greater  than  —  4. 

b.  Less  than  —  5.  e.    Numerically  greater  than  6. 

c.  Greater  than  —  4.  /.    Numerically  less  than  —  5. 

9.  7.  12.    -18.  15.    -3.  18.  5. 

10.  -10.  13.    -    1.  16.        2.  19.    -    6. 

11.  -    8.  14.  0.  17.    -4.  20.    -    |. 

25.    State  the  absolute  value   of   each   of  the   numbers   in 
Exercises  9-20. 


42  A  HIGH   SCHOOL   ALGEBRA 

WRITTEN   EXERCISES 

Determine  which  is  the  greater  in  each  of  the  following 
pairs  of  numbers,  and  write  the  relation  by  use  of  the  sign  >  : 

1.  8,  6.  4.    -6,5.  7.   0,  10. 

2.  3,  4.  5.    -  6,  -  5.  8.   0,  - 10. 


3.    -  5,  6.                 6.    6,  -  9. 

9.    -4,  -2. 

80.   Processes  with  inequalities. 

1.   Addition : 

(1)        4>3 

(2)            3  <  4 

5>2 

-5<  -2 

Adding,  4+5  >  3+2  Adding,  3-5  <  4-2 

TJiat  is,  if  tivo  inequoMties  of  the  same  kind  are  added  the 
resMt  ivill  be  an  inequality  of  the  same  kind,  but  the  sum  of  two 
inequalities  of  different  kinds  may  result  in  an  equality  or  in  an 
inequality  of  either  kind.  Similarly,  the  subtraction  of  inequali- 
ties is  uncertain. 


2.   Multiplication: 

(1)                   3  <  5 

then        2  .  3  <  2  .  5 

(2)               _  2  >  -  5 

then    3(-2)>3(-5) 

If  the  members  of  an  inequality  be  midtiplied  by  any  number  not 
zero  or  negative,  the  result  icill  be  an  inequality  of  the  same  kind. 
If  the  multiplier  is  negative,  the  result  will  be  an  inequality  of  the 
opposite  kind. 

ORAL  EXERCISES 

Add: 

1.   3>2  2.       3  >2  3.    -3<  -2 

4>3  -4> -5  -7< -5 

4.  Multiply :  5  >  2  by  3 ;  also  by  —  2;  also  by  —  1. 

5.  Multiply :  a  <  6  by  3 ;  also  by  —  2 ;  also  by  —  1. 


RELATIVE  NUMBERS  43 

REVIEW 
ORAL  EXERCISES 

1.  The  temperature  was   —  8°  at  6  o'clock  and   +  5°  at 
9  o'clock.     How  many  degrees  did  it  rise  in  this  interval  ? 

2.  A  ship  sailed  on  a  meridian  from  Lat.  -f  12°  to  Lat.  —  2°. 
Through  how  many  degrees  did  it  sail  ? 

Read  in  full : 

3.  11  +  18.  6.  3 +(-2).  9.  xy-(-xy). 

4.  14-9.  7.  p^(-q).  10.  ah -{-ah). 

5.  _2-(+3).  8.  -3a--(H-26).  11.  m7i  +  (-2m). 

Add: 

12.    -6  14.    -16  16.    -    8  18.    -18 

-9  7  26  36 


13.      15 

15. 

23 

17. 

40 

19. 

33 

-9 

-9 

-15 

-13 

Subtract : 

20.        9 

22. 

-15 

24. 

-14 

26. 

8 

-5 

-   6 

8 

-16 

21.    -8 

23. 

-32 

25. 

-   3 

27. 

-14 

4 

-    3 

15 

-14 

Multiply : 
28.         6 

30. 

-4 

32. 

-6 

34. 

-12 

-5 

-8 

10 

3 

29.    --8  31.    -15  33.    -11  35.    -10 

3  3  -    3  0 

Divide : 

36.  -  15  by  3.  38.   18  by  -  6. 

37.  -  20  by  -  4.  39.    40  by  -  10. 


44  A   HIGH   SCHOOL  ALGIiiBRA 

WRITTEN   EXERCISES 

1.  Write  the  sum  of  positive  a  and  negative  6. 
Indicate,  by  using  the  signs  +,  —  : 

2.  $15  lost  plus  $10  gained. 

3.  The  sum  of  positive  x  and  negative  y. 

4.  The  sum  of  negative  m  and  positive  n. 

5.  The  difference  of  positive  m  and  negative  n. 

6.  The  difference  of  positive  x  and  negative  y. 
Perform  the  indicated  operations : 

7.  _6  +  5-4(-3).  10.   16---2-(4. -3). 

8.  9-3(-3)+6.0.  11.   12. -3+24- (6. -3). 

9.  15^-54-(3. -5).  12.    17(2  -  8)- (70 -i-- 7). 

13.  (-5.15-3)--(15-4.6). 

14.  _6a-2(-4a)-(8a^-2). 

15.  Multiply  —2x>  —  5xhy  —  3. 

SUMMARY 
The  following  questions  summarize  the  definitions  and  pro- 
cesses treated  in  this  chapter : 

1.  Wh?it  are  relative  numbers  ?  Negative  numbers'^     Sec.  56. 

2.  What  signs  are  used  to  indicate  the  positive  and  negative 
character  of  numbers  respectively  ?  Sec.  59. 

3.  State  the   rules   that  determine  when  the  signs  +,  — , 
indicate  signs  of  operation.  Sec.  59. 

4.  Why  are  the  numbers  of  algebra  called  signed  numbers  ? 

Sec.  60. 

5.  Define  absolute,  or  numerical  value.  Sec.  61. 

6.  If  the  addends  have  like  signs,  what  is  the  sign  of  their 
sum  ?  Sec.  63. 

7.  In  adding  two  numbers  with  unlike   signs,  how  is  the 
sign  of  the  sum  found  ?  Sec.  67. 

8.  How  are  algebraic  numbers  subtracted  ?  Sees.  70,  71. 

9.  State  the  law  of  signs  in  multiplication.  Sec.  75. 


RELATIVE  NUMBERS  45 

10.  State  the  law  of  signs  in  division.  Sec.  77. 

11.  Define  the  greater  of  two  numbers  ;  the  less.         Sec.  78. 

12.  De^ne  numerically  greater ;  algebraically  greater.  Sec.  79. 

13.  What  is  the  result  of  adding  inequalities  of  the  same 
kind  ?     Of  different  kind  ?  Sec.  80. 

14.  What  is  the  result  of  multiplying  both  members  of  an 
inequality  by  a  positive  number  ?     By  a  negative  number  ? 

HISTORICAL  NOTE 

For  the  first  use  of  negative  number  we  must  turn  to  the  Brahmin 
schools  of  India.  The  Hindoo  priests  were  clever  mathematicians,  and 
tradition  relates  that  the  great  reformer,  Buddha,  in  his  youth  won  the 
maiden  he  loved  by  solving  a  difficult  set  of  problems.  Hindoo  scholars 
did  much  to  develop  algebra,  and  the  writings  of  Bhaskara,  who  lived 
about  1150  A.D.,  is  a  summary  of  their  work.  The  poetic  tendency  of  the 
Hindoos  affected  all  their  thinking,  and  they  expressed  their  mathematics 
in  flowery  language  and  in  verse.  Bhaskara  called  one  part  of  his  work 
Lilavati  ("noble  science  "),  and  proposed  many  problems  like  the  follow- 
ing: "  The  square  root  of  half  the  number  of  bees  in  a  swarm  flew  to  a 
jasmine  bush  ;  |  of  the  whole  swarm  remained  behind  ;  one  bee,  allured 
by  the  sweet  odor  of  a  lotus  flower,  became  entangled  in  it  while  his 
excited  mate  lingered  abo\it.  Tell  me  the  number  of  bees. "    (Answer,  72.) 

The  Hindoos  were  the  first  to  explain  positive  and  negative  numbers  by 
reference  to  debits  and  credits,  and  the  modern  interpretation  of  these 
numbers  by  opposite  directions  on  a  straight  line  was  not  unknown  to 
them.  They  discovered  that  the  solution  of  certain  equations  gave  negative 
roots,  but,  strange  as  it  may  seem,  they  rejected  them  and  negative  numbers 
received  no  further  acceptance  until  Descartes  fully  interpreted  them  (1637  ) . 
They  had  so  long  remained  a  mystery  that  they  were  known  as  "absurd  " 
or  "fictitious"  numbers.  Even  as  late  as  1545  Cardan  called  them 
numerce  fictae.  Hence,  we  may  in  a  measure  be  forgiven  for  calling 
negative  number  artificial  in  comparison  with  positive  number.  It  is 
unfortunate,  however,  that  the  negative  number  came  down  to  us  labeled 
"  artificial "  or  "unreal "  number,  because  it  is  just  as  real  as  the  positive 
number.  Both  kinds  have  to  exist,  or  neither  could  exist.  Both  positive 
and  negative  numbers  are  concrete.  They  are  denominate,  or  named  num- 
bers, just  as  much  as  3  feet,  or  7  quarts.  It  is  quite  as  natural  to  think 
three  positive  and  three  negative  units  as  it  is  to  think  three  dollars'  gain 
and  three  dollars'  loss.  When  the  unit,  negative  one,  has  been  defined 
to  be  an  absolute  one  taken  in  the  opposite  sense  to  positive  one,  we  can 
as  easily  count  or  reckon  with  negative  numbers  as  with  positive  numbers. 


CHAPTER   V 
ADDITION 

ADDITION    OF   MONOMIALS 

81.  Algebraic  Sum.     The  result  of  adding  numbers  some  or 
all  of  which  are  negative  is  called  their  algebraic  sum. 

82.  Like  Terms.     Terms  or  monomials  that  have  the  same 
literal  factors,  are  called  like  terms  or  like  monomials. 

For  example,  the  following  are  pairs  of  like  terms  : 

ab  and  ab  ;    6  a  and  —3a;    4  a6  and  ^ab;    2  a^b  and  \  a^b. 

Like  terms  in  algebra  correspond  to  numbers  in  arithmetic  having  a 
common  factor. 

For  example,  5x7  and  3x7;  or  7  x  15  and  6  x  15, 

ORAL  EXERCISES 

From  the  following  list  select  terms  like  the  first ;  like  the 
second ;  like  the  third : 


1. 

ah^x. 

5. 

2  6  V. 

9.    —4  c/. 

2. 

cf. 

6. 

amp. 

10.    -6iab\ 

3. 

ah\ 

7. 

-iab\ 

11.   ax. 

4. 

mp. 

8. 

5  ab'x. 

12.   24  c/. 

83.    Terms  are  alike  in  any  letter  if  they  contain  the  same 
power  of  that  letter. 

Thus,  2  ax^  and  bx^  are  alike  in  x. 

Which  of  the  terms  of  Exercises  1-12  are  alike  in  a  ?   In  6  ? 
In  a;? 

46 


ADDITION  47 

84.  To  add  two  like  terms  or  monomials,  prefix  to  their  like 
factor  the  sum  of  their  coefficients. 

In  arithmetic  we  add  3x7  and  5  x  7  by  adding  3  and  5  and  writing 
8x7. 

Similarly,  mSax  +  5ax  =  S  ax,  we  add  3  and  5  and  write  8  ax. 

In  —  4  a62  +  6  ab'^  the  like  factor  is  ab^  ;  hence,  adding  the  coefficients 
—  4  and  6  the  whole  sum  is  +  2  ab^. 

The  addition  of  dissimilar  terms  can  only  be  indicated. 


Add 


1. 

-5x 

3.        82/ 

-2x 

-52/ 

2. 

-6x 

4.    Ga'b 

2x 

2a''b 

ORAL  EXERCISES 


5. 


-15  2/2 

7. 

-6a26 

- 10  2/2 

-  3  a'^b 

-lOxy 

8. 

16  mn 

5xy 

—    9  mn 

9.    —7ab-h(-Sab)  =  ()ab.     11.    .7  x-\- (- .3  x)  =  ()x. 
10.   Sabc-\-(-5abc)=()abc.      12.    S  a^ -^  (-5  a^)  =  {  )aK 

85.  To  add  several  like  terms,  combine  them  in  order,  or  add 
the  positive  and  the  negative  terms  separately,  and  then  combine 
these  two  sums. 

o  a  Thus,    in    the    column    on    the    left,    adding    upward, 

—  5a      —  Qa +  12  a  =  +  6a,    then    6a  +  (—5a)  =  a,    and   finally 
12  a      a  +  3a  =  4a;or  the  sum  of  the  positive  terms  is  15  a,  the 

—  Q  a      sum  of  the  negative  terms  is  —  11  a,  and  the  sum  of  these 
two  is  4  a. 


4a 


Add 


WRITTEN   EXERCISES 


76 

2. 

12  y 

3. 

4  m^ 

4. 

-2x 

5.        Sxy^ 

-36 

-92/ 

—  m^ 

Sx 

10  xy' 

20  6 

5y 

25  m^ 

-9x 

-  20  xy^ 

48  A  HIGH   SCHOOL   ALGEBRA 


6. 

4s 

7. 

-x'y 

8. 

76 

9. 

f 

10. 

20  w 

-12  s 

dx'^y 

86 

2f 

30  w 

6s 

-2x'y 

-20  b 

-5f 

-19w; 

-9s 

^x'y 

3b 

25  f 

11  10 

11.   3x-Sx  +  15x=?         12.    6x  +  Sx-3x-4.x=? 

13.  ax''-^ax'^  +  la3i^  =  (  )  ax\ 

14.  ^  a.62  +  1.  a62  _  4  ^52  _^  I  ^,52  ^  (  ^^^2^ 

15.  -Ux  +  23x-]-99x=? 

16.  40  a -75  a +  89  a  =  ? 

17.  12  ab  —  IS  ab  i- 75  ab=? 

18.  13  a;?/ -50  0^2/ +  113^^=? 

19.  -  0^2  +  4  x2  -  10  ar^  =  ? 

20.  —xy  —  4:0  xy  -\-9  xy  =  ? 

21.  —  m?i^  +  12  mn^  — 15  mri^  =  ? 
"  22.  15  ic?/2  +  33  xy^  -  48  xif  =  ? 

23.  422-52^2 _|.io9;22_i222  =  ? 

24.  29aj2_43^2_^37^^j^9 

25.  —  20pg  — 100p^  +  7pg  =  ? 

26.  -  2/2  +  4  ?/2  -  117  2/2  +  3  2/'  =  ? 

27.  -  12  i9g?'2  _  17  pg^2  _  53  pg^2  ^  9 

Add  the  terms  in  x  and  the  other  terms  separately : 

28.  17  a;  — 5a;-h3cc  +  2  — 11»  — 4a;-}-10a^  — 27. 

29.  22  a;  —  3  a^  —  6  —  4  a;  4-  5  a;  —  a;  —  2  a;  —  9. 

30.  2  a;  +  3  a  + 15  a;  -  12  a;  +  6  a  -  24  a. 

ADDITION   OF  POLYNOMIALS 

86.    The  addition  of  polynomials  is  similar  to  that  of  de- 
nominate numbers. 

For  example : 

Denominate  Numbers  Polynomials 

Just  as :          3  bu.  4  qt.  so :       3  6  +  4  g 

plus      5  bu.  3  qt,  plus      bb  -\-  3q 

equals  8  bu.  7  qt.  equals  8  6  +  7  g 


ADDITION  49 

ORAL  EXERCISES 

1.   Add:  4mi.  3rd.  2  ft.  also,  4m  +  3r  +  2/ 

6  mi.  7  rd.  8  ft.  ,  6m-\-7r  +  Sf 

State  the  numbers  to  fill  the  blanks  in  the  following  addi- 
tions : 

2.  3. 

2a-f  46  2a4-      &+     c 

3a+  55  5a+  36+  2c 

()«  +  ()&  ()«  +  ()^  +  ()c 

87.  To  Add  Polynomials :  ^rra^ipfe  the  like  terms  in  columns 
and  add  as  in  the  case  of  monomials,  using  the  signs  obtained  as 
the  signs  of  the  result. 

For  example  : 

Not  Arranged  Arranged 

a  +  c  +  6  a-\-    h-\-c 

—  3  6  +  g  H-  c  g  — 36+c 

Here  the  first  column  is  4-a  +  a  =  +  2a;  the  second  column  is 
-f6  —  36=  —  2&;  the  third  column  is  +  c  +  c  =  +  2  c.  The  terms  thus 
obtained  with  their  signs,  namely,  2a  —  26  +  2  c,  constitute  the  sum  of 
the  polynomials. 

88.  Commutative  Law  of  Addition.  Tlie  sum  of  two  or  more 
terms  of  a  jwlynoinial  is  the  same  in  whatever  order  the  terms  are 
taken. 

The  rearrangement  of  the  terms  of  polynomials  before  adding  is  an 
application  of  this  law. 


WRITTEN   EXERCISES 


Add: 

1.    5a-3  6 

3.     ^X-^Ly 

5.    ip  +  Uq 

o-    6 

y  +  ^x 

i^+    |i> 

2.    2a~6c 

4.    3m-l.ln 

6.    —7x-\-   4  2/ 

3c4-2g 

6m-    .9n 

9x-10y 

50  A  HIGH  SCHOOL  ALGEBRA 


7. 

X^J^2f-       ;,2 

12. 

17p2_      ^^4-^2 

22_3^2^2a^5 

-6p2  4.2p^_.92 

8. 

at^-^at  ^    c 

13. 

2a  +  764-llc 

2at  -Sat^-{-2c 

26+    9c 

9. 

rn? -\-    m  +1 

14. 

12«+  8  2/+lT« 

m  -  2  m2  -  8 

9a;+122;+132/ 

10. 

i>'+    P  +  8 

15. 

iP+   |g  +  f^ 

p2  4-9^4-6 

7ip  +  9Jg+    r 

11. 

a?-\-Q>x'^—   9  a; 

16. 

ir-   |s4-4« 

2a?  +  ^x''-l(ix 

|r4-Hs-i« 

89.  Preparatory. 

Find  the  value  of  each  expression  when  each  letter  =  1 : 

1.  a +  2  6.  4.   2  a  — 2  6.  7.   3  a  — c. 

2.  a  +  6  +  c.  5.    6+2  c- a.  8.   a  +  d+3c. 

3.  c+4fZ-2a.         6.   2a  +  36+3c.       9.   36  +  c  +  a. 

90.  Test  of  Addition.  To  test  the  work  of  addition,  substi- 
tute unity  for  the  letters.  The  value  of  the  sum  must  equal 
the  sum  of  the  values  of  the  expressions. 

In  practice  the  work  and  test  are  written  as  follows : 

Solution  "Test 

2a-5b  -3 

4 a +4 6  +8 

ea-    b  +5 

The  use  of  unity  tests  the  coefificients  including  their  signs,  but  does 
not  check  mistakes  made  in  writing  the  literal  parts.  Such  errors,  how- 
ever, rarely  occur,  and  are  easily  discovered  by  inspection.  Numbers 
other  than  unity  are  apt  to  make  the  work  of  checking  too  complicated. 

WRITTEN   EXERCISES 

Add  and  test : 

1.      a+   3  6  2.    6c+     d  3.    12^-    61^2 

lla  +  106  3c  +  2d  +  e  8^  +  12^2 


ADDITION  51 

4.   4.x        +z  8.    12a-\-5b  12.   4.0-^gt^ 

2z+y  +  x  6a-Sb  50  +  f  ^^^ 

13.  45  m  —  2  w  4-  g 
'5  m  +3ri-|-  g 

14.  3  a2  -  5  a  + 1 
4a  4.8a2_3 

15.  5  a;  4-     2/~     ^ 
Sx  —  7y  +  Sz 

16.  ia:4-    i2/+    .9  2;  18.       a;2+       a!2/  +  3?/2 
|a;  +  l|y  +  l.lg  6  x"^ -\- 10  xy -\- 5  y^ 

17.  1.1  a -8.9  6+    c  19.   x'^-{-    x^y-\-Sy^ 

3.9  a+    .56 -5c  4  x^y  +    y^  +  g'^ 

20.        5  a—    7  6  23.                 p+   3g 

3a  +  10  6  m  +  3p+      g+r 

-6a  +  186  5m-\-2p            +6r 

-7a  +  12b  10g  +  5r 


5. 

40  m+       n 
5m-S9n 

6. 

17  a+   4  6 
3  a  -  16  6 

7. 

6a  +  96 
4a-     6 

9. 

-15a;  +  12 
-    8a;-  3 

10. 

5a;2  +  3x 
-  3  a;  +  7  a;2 

11, 

4.xy-  2z^ 
5xy-\-10z' 

21.            x"--   5x 

24. 

a2+   4      -      a 

-    4a;2+   3  a; 

5     -   3a2+   2a 

-12a;2+       X 

6a^-\-   8a  -   5 

15a;2-12x 

4:a  -    7a2-   2 

-10a:2  +  16a; 

8  a  - 12      -  15  a2 

22.       a  +  2  64-    3& 

25. 

4<7+   3v  — 7a; 

2  a           +  66  c2 

5a.'+   22/  — 4^; 

9  6+      & 

2y—    Sg—7v 

a  +     6  +       e 

13v-lla;  +  2^ 

5a-    b 

12x-15g  +  4.y 

26.  A  dealer  bought  at  one  time  3  kinds  of  coal,  50  a  tons 
of  the  first  kind,  10  6  tons  of  the  second  kind,  and  12  c  tons  of 
the  third ;  at  another  time  he  bought  75  a  tons,  15  6  tons,  10  c 
tons  respectively  of  the  same  kinds.  How  many  tons  did  he 
buy  in  all  ? 


52  A   HIGH   SCHOOL  ALGEBRA 


Add 
1. 


REVIEW 
ORA-L  EXERCISES 


—  6ab  6.   4  7rr2  11.    -9pq^ 

—  S  ab  2  7rr^  —  7  jjq"^ 

2.  12  ac2  7.    12a;?/2  12.   i  tt/-^ 

—  8  ac'^  IS  xyz  f  7rr^ 

3.  mr^  '  8.    —     mpq  13.    —  6  a^6 

—  mr^  —6  mpq  8  a^6 

4.  i  mv^  d.    —  ^  mv^  14.        4  xy"^ 

—  mi>^  —  2m'?;'^  —  7  ic?/^ 

10.        43^u^  15.        16iiv^ 

-23w^  -    9wf2 

20  w3  _    J^^^2 

—  10  w)^  25  1*^2 

16.  7a;2-2a;-5  18.   a' -{- 3  a'b  +  S  ab' -\- b' 
4a;2_^4a._l_5  g^  -  3  a^6  +  3  a^j'  -  b^ 

17.  4.5  m +  3.2  71+     p  iq,   Sx —  Ay -^7  z —  9 

.5  m +.lj9  8?/  —  2g  +  4a;  —  3 

WRITTEN   EXERCISES 

Add  and  test : 

1.  a-j-b-{-c,   2a  +  6  +  3c,   a  +  &,    6b-i-5c. 

2.  10  a  +  9  ^>  +  c,    9  a  +  10  c,    I  a  +  i  6,  2  5  +  c. 

3.  .9  07  +  .3  2/  +  2;,    .1  x-\-  .7  y,   5  y  -\-^z,   4:X-j-z. 

4.  3a-2?>,   4rt+76,    -6a-b,   14  a -21  6. 

5.  m^+bmp,    7m'^—Smp,    —12m'^+32i',   6m'  — Amp -{-7 p\ 

6.  5ax  +  b,   2  ax  — 3  b,    —Sax+7b,    —  20  ax  —  ISb. 

7.  ia;-i2/,    -fa^  +  i2/j    •'^^  +  |2/j    -.3a;  +  |2/. 

8.  7  +  2a;2,    3a;2-l,    4  -  5  a;2. 

9.  4a  +  7  6,  l^a-6c,   3c  +  5  6,   46-7a. 
10.  a;2  +  7a;-4,   3a;2-5a;,   4a;2-lla;  +  2. 


ADDITION  53 

11.  5^-3  +  7^^   ^-3^^   f  +  9t^-15-\-St. 

12.  3a6  +  7ac,   5ac—2bc,   6bc+9ab,   8  6c  — 18 ac. 

13.  x  +  A,   x'^—.5,   3a;— .7,   x'^  —  .9  +  4:X. 

14.  2y'-4.y-\-y'-l,   Sy -f -^Sy^ -15,   Sy-7  +  lly* 

-15y%  42/3  _^  12  2/2-6  +  2/4,   ll-y^^f_Sy\ 

15.  A  grocer  had  7  a  dollars  on  hand ;  his  ten  salesmen 
took  in  4:  a,  5  G,  2  b,  6  a,  S  a,  7  c,  4  6,  2  c,  5  c,  11  b,  dollars,  respec- 
tively.    How  much  had  he  then  ? 

16.  A  merchant  made  the  following  bank  deposits:  On 
Monday  3  a  dollars  in  gold,  4  b  dollars  in  silver,  and  9  c  dollars 
in  bills;  on  Tuesday* a  dollars  in  gold  and  15c  dollars  in 
bills ;  on  Wednesday  b  dollars  in  silver  and  12  c  dollars  in 
bills.     How  much  did  he  deposit  all  together  ? 

SUMMARY 

The  following  questions  summarize  the  definitions  and  pro- 
cesses of  this  chapter : 

1.  What  is  meant  by  algebraic  sum?  Sec.  81. 

2.  What  are  like  terms  or  monomials?  Sec.  82. 

3.  How  may  like  terms  be  added  ?  Sees.  84,  85. 

4.  How  may  polynomials  be  added  ?  Sec.  87. 
6.  State  the  Commutative  Laiv  of  Addition.  Sec.  88. 
6.   Explain  a  test  for  the  work  of  addition.  Sec.  90. 


CHAPTER  VI 
SUBTRACTION 

SUBTRACTION  OF  MONOMIALS 

91.  Subtraction  is  the  process  of  finding  the  difference  be- 
tween two  numbers,  called  the  minuend  and  the  subtrahend. 

92.  Difference.     The  difference  is  the  number  which  added 
to  the  subtrahend  makes  the  minuend.     Sec.  69. 

ORAL  EXERCISES 

State  the  differences  of  the  following : 


1. 

20  a 

4. 

21aj2 

7. 

90m 

10. 

40  mn 

15  a 

14a;2 

45  pg 

39  mn 

2. 

47  6 

5. 

17  6c 

8. 

^Ozio 

11. 

54.  d 

27  6 

9  6c 

25  zw 

24:  d 

3. 

12  62c 

6. 

39  c2 

9. 

30  xy 

12. 

19a6a; 

7  6^0 

19  c2 

17  xy 

12  a6a; 

93.    Since  subtraction  is  the  reverse  of  addition,  we  can  sub- 
tract a  number  by  adding  its  opposite. 

For  example,  "  to  subtract  3  "  and  "  to  add—  3  "  mean  the  same  thing. 
Likewise  "  to  subtract  —  5a"  and  "  to  add  6a"  mean  the  same  thing. 

To  subtract  one  number  from  another,  change  the  sign  of  the 
subtrahend  and  add  the  result  to  the  minuend. 

Thus,  to  subtract  _  ^^^  change  to  ^^^  and  add. 

The  pupil  should  learn  to  make  the  change  of  sign  mentally. 

54 


SUBTRACTION 


55 


ORAL  EXERCISES 


Find  the  differences : 


1. 

18 

-99 

2. 

6ab 

6  ah 

3. 


5. 


7. 


ox 
15x 


5xy 
— 10  xy 

—  Sabc 
Sabc 

-12  m 
8  m 


8. 


9. 


6  m^'^ 
19  mv^ 


10. 


11.   5a 

8a 


-1.5  s 

-3.5  s 

—   4r^W 

-9iw 

12. 


13 


4:X 

1  X 


12  y 

14.    —Sx 

2x 


15. 


16 


18. 


19. 


20. 


21. 


8a; 


Uy 

-  ^y 

17.    16  6 
116 


-Ud 

—  Id 

—  4p 
-7p 

2  m' 
-2  m' 


29.  46  —  52.  33. 

30.  4  a  — 7a.  34. 

31.  8a;2_iQ^^  35 

32.  8a6  — 15a6.  36.    ttt' 


40  m  —  46  m.  37. 
13  pq  — 15  pq.  38. 
4-  mv'  —  mv-.          39. 


4  7rr2. 


40. 


22. 


23. 


24. 


25. 


26. 


27. 


28. 


1-^ 

10/s 
45  r^ 


2m2 

2  m' 

23  a 
—  6a 

23  a 
6  a 

-23  a 
6  a 


-23  a 

-   6a 

23  a:* 

-    6a;* 

-    t 

,-3t 

-2  7rr3. 

-  15/s. 
-50r^. 


SUBTRACTION   OF  POLYNOMIALS 

94.   The  subtraction  of  pol5momials  is   similar   to  the  sub- 
traction of  denominate  numbers. 
For  example : 

Denominate  Numbers  Polynomials 

Just  as :  5  lb.  4  oz.  so :  5  Z  +  4  2r 

minus  3  lb.  3  oz.  minus  3  Z  +  3  0 

equals  2  lb.  1  oz.  equals  2  Z  +  1 0 


Subtract : 
1.   12  bu.  3  pk.  7  qt. 
9  bu.  2  pk.  4  qt. 
5 


ORAL  EXERCISES 


2.    12  6 +3^5 -I- 7  g 
964-2p  +  4g 


56  A  HIGH   SCHOOL   ALGEBRA 

State  the  numbers  to  fill  the  blanks  in  the  following : 

3.    6xy-\-Sy^  4.   5a  +  3  6^  4.  8c 

J_xji±2f_  g  4-     b'-\-5G 

O-^y  +  Of  ()  +  ()  +  () 

Subtract : 

5.  6  a -{-5b  S.   20x-^5y+z 

a4-36  9x-{-  5y 

6.  16m+2?i  9.  45a  4-3  6  + 12c 

12m  4-    n  5a-\-    6  +    2c 

7.  17c  +  9cZ  10.    10i^_|_g^2_^5^2 
17  c  4-     d  5a^4-    ^2+5^2 

95.  To  subtract  a  polynomial  arrange  its  terms  under  the  like 
terms  of  the  minuend,  subtract  each  term  from  the  one  above  it, 
and  use  the  signs  obtained  as  the  signs  of  the  result. 

In   the    example,   the    first    column  is  2  a  —  a  =  a ; 
2  a  _  2  6  +     c    the    second    column    is    -  2  &  —  0  =  -  2  6 ;     the    third 
g  +     0  -  2c    column    is     +c-(— 2c)  =  +3c.      The     terms    thus 
fj  _  2  &  +  3  c    obtained  with  their  signs  constitute  a  —  2  &  +  3  c,  the  dif- 
ference of  the  polynomials.     When   either  polynomial 
lacks  a  term  to  correspond  to  a  term  of  the  other  polynomial,  supply  zero 
in  its  place. 

96.  Test  of  Subtraction.  Use  arbitrary  values  to  test  sub 
traction.  The  sum  of  the  values  of  the  difference  and  the 
subtrahend  must  equal  the  value  of  the  minuend. 

Solution  Test  :  Let  each  letter  =  1 

3cc-4y  +    c  3-44-1  =      0 

x+     y-Sc  1  +  1-3:^  -1 

2x-52/  +  4c  2-544=41 

In  practice  it  is  sufficient  to  write  the  following  : 

Solution  Test 

Sx  —  iyi-     c  0 

x+     y-Sc  -  1 

2x- 6y 4  4c  1 


SUBTRACTION 

WRITTEN   EXERCISES 

Subtract  and  test 

1.   Sa-\-b 

3.    ia-ib 

5.    ip  +  iq 

a  —  b 

a  +  |6 

iP-iQ 

2.    6a -36 

4.   3  m -.In 

6.    —7  a; +  4  2/ 

5a +  7  6 

6m-.9n 

10a;-9?/ 

67 


Arrange  the  like  terms  in  columns  and  subtract  : 

7.  Sx  +  12q-\-6a  9.    5c  +  66-f  10 a 
4a+    5ar +3g  5+5a+    5c 

8.  j9+3n  +  45m  10.    10aa;+    mp  +  5pq 
5  m  +  2n-\-p  2pq  -\-  \ mp -\-  5  ax 

11.  3a;2/—       s;^  15.   2  a  +  c  —  2  &        19.   ^  x^ -\- 2  xhf -\- z^ 
5  .Ty  +  10  g^  a  +  6-2c  a;^  +  2  x'y'- 

12.  12?-    6^2  16.    7aj2_2a;  +  4       20.  4  a; -3?/ +  8 

9  ?  -  12  ^'^  2  a;^  +  3  a;  -  1  2y  +  5z-l 

13.  40-i^?2  lY^   2z-\-lx  +  y        21.   4a4-26-9 
50-1  gf^^  2x+     y-z  8c  +  4a-6(^ 

14.  5aj  +  7?/-82;       18.    6c  +  3d  +  e        22.    2x^  +  505-1 
3a;+    y-4g  3  c  +  2  (^  3  x-^ -  7 a;^  +  8  a? 

23.  A  broker  had  7  a  +  5  6  dollars  in  a  bank  and  withdrew 
a  +  4  &  dollars.     How  much  did  he  still  have  in  the  bank  ? 

24.  A  coal  dealer  bought  c  carloads  of  coal  containing  40  tons 
each,  and  d  carloads  of  50  tons  each ;  he  sold  5  c  tons  to  one 
customer,  8  d  tons  to  another,  and  12  c  tons  to  another.  How 
many  tons  had  he  left  ? 

97.   Removal  of  Parentheses. 

1.  If  a  parenthesis  is  preceded  by  the  sign  +?  the  terms 
within  the  parenthesis  are  to  be  added  to  what  precedes,  hence 
the  parenthesis  may  be  removed  without  altering  the  value  of 
the  expression. 

For  example  :  a  +  (&  +  c)  =  a  +  &  +  c. 

a -\-{b  —  c)=  a +  h  —  c. 
a  -{-(—  b  —  c)=  a  —  b  —  c. 


58 


A  HIGH  SCHOOL  ALGEBRA 


2.  If  a  parenthesis  is  preceded  by  the  sign  — ,  the  terms 
within  the  parenthesis  are  to  be  subtracted  from  what  pre- 
cedes; hence  the  parenthesis  may  be  removed  provided  the 
sign  of  each  term  within  the  parenthesis  is  changed,  each  sign 
+  to  the  sign  — ,  and  each  sign  —  to  the  sign  -f .  Sec.  93. 


For  example : 


a  —  (6+  c)=  a  —  6  —  c. 
a  —(b  —  c)=  a  —  b  +  c. 
a  —  {—  b  —  c)  =  a  +  b  +  c. 


WRITTEN   EXERCISES 

Remove  parentheses  and  unite  terms  when  possible : 


1.  a+(b-Sa). 

2.  o-(«  +  2). 

3.  3a-{2a  +  b). 

4.  76-(4a-6&). 

6.  a -(25 -5a) -4  6. 

6.  11  ^+(-3^-1). 

7.  4.x  -}-7y-(3x-\-2y). 

8.  d-\-3d''-(2d-d''). 

9.  5-3p-f(-18+2p). 
10.  a-bx-(2a  +  bx). 

21.  4m3-7m2  +  3m-(- 


11.  7q  +  5-(-ll-3q). 

12.  -(-3x-2tj-{-llz). 

13.  _(4a  +  3  6-6c). 

14.  a5-3a;2+7-(2a^2_^5-3a;). 

15.  9^  +  3 -(2^-1). 

16.  5  a; -12  2/ -(3  a;  4-2?/). 

17.  4.a  +  7b-(2a  +  3b). 

18.  7  -  6  m  -  (1  4-  3  m). 

19.  lli)  +  l-(-i)  +  3). 

20.  15x''  +  Tx-{lSx^-3x). 
2  m^  -  9  m2). 


22.  a"^  -  5  ab  -\-  7  ac  +  b'^  -  (Aab  -{-  7  a^  -  6  ¥). 

23.  x^ -\- 3  X2J  ~  4:  xz  -{- 7  y^  —  (z"^  —  3xz  —  4:X^—  y^). 

24.  Calculate  the  value  of  A—(2B  —  C)  when 


(1) 

(2) 

(3) 

(4) 

(5) 

(6) 

(7) 

^  = 

24 

1 

-4 

X 

4d 

a+     X 

i>  +  3 

B  = 

3 

-5 

3 

2x 

-6d 

3a-     X 

g  +  5 

0  = 

8 

9 

-5 

3x 

3d 

4.a-\-7  X 

2i)  +  3g 

SUBTRACTION  59 

98.  The  methods  of  Sec.  97  may  be  applied  when  there  is 
a  parenthesis  within  a  parenthesis.  In  this  case  the  signs, 
5  I,  [  ],  ,  are  commonly  used  to  distinguish  the  different 

parentheses.  They  may  be  removed  one  at  a  time,  usually  the 
inner  one  first,  although  it  is  likewise  possible  to  begin  with 
the  outer  one. 

For  example : 

Beginning  with  the  inner  parenthesis, 

5a-{6a  +  36-(2a-5  h)}=  5a-{6a  +  36-2a  +  6  6} 
=  5  a  -{4  a  +  Sb] 
=5a-4a-86 
=  a-8&. 

Beginning  with  the  outer  parenthesis, 

6a-{6a  +  3b-{2a-6b)}=5a-6a-Sb+{2a-6b) 
=  -a-Sb  +  2a-5b 
=  a-Sb. 

When  the  first  parenthesis  is  removed  the  signs  within  the  second  one 
are  not  changed  because  the  expression  is  taken  as  a  single  term. 

WRITTEN    EXERCISES 

Eemove  parentheses  and  unite  terms  as  much  as  possible : 

1.  4:X-\3x-(2-i-x)l  4.    a;2_j3^2_(2a;2  +  l)|. 

2.  7-h\4.-(5x-^-2)  +  3xl.    5.    -  \4:x''-^(3  x~  5x^ -9x)l. 

3.  2  +  (5a-3a  +  4a)  +  6a.    6.    a-3b—{b-3a  +  (3b-a)\. 

7.  p+(3p-q)-{3q-p)+q. 

8.  a2-(62_c2  +  a2)-(c2-62). 

9.  3x7j-^2y^-(x'^-xy-{-2y^). 

10.  3a  +  b-lG-ld-3a-\-b-c']-{-d\. 

11.  m—  j3 m  — [4 m—(5 7n  —  6m  —  7m)  —  8 m]  +9 m\  +10 m. 

12.  5a-[2x-\4.c+(7  c-x)-3c]  +  4:a']. 

13.  a^-[a'-(l-a)]-[l-]-\a''-(l~a)-\-a'iy 

14.  6  m—  \p--{-  2  q  —  (m  -\- q)-\-'3  —  (5 p  —  3  q -  4:  m)\. 

15.  ab  —  ac  —  \2ab—(3ac-{-bc)  —  2bc-\-2ac-(ab  —  2ac)\, 

16.  b-lx-\3  +  3b-{5x  +  2)\-{4.x-3b-l)2' 


60  A   HIGH   SCHOOL   ALGEBRA 

17.  m+p-[-[2m  +  2p-(g  +  r)]-[4p 

—  (r  — g  +  m)— p]  +  3m|. 

18.  2a  +  6+(12a-4  6)-j3a  +  3  6-(5a-3a-3  6)|. 

19.  ^a-:^x-(la-}x)-(2n-ix-\a)-^^a. 

20.  3-S[42/-(3-^^r2)]_[;a+(52/-^T5)]J. 

21.  xy  —  l3-\-a  —  (x  +  z  —  xy-^a)']-{-la—(y  —  z  —  5)]. 

22.  -  [3  a6  -  J5  6c  -  3  d|  +  (^  - (4  6c  +  f  2  (^  +  3  a6 J  H-  d)]. 

23.  x-(y-z)-\x-[y-z-(y-{-z-x)+(x-y)]l. 

24.  l-y-il-y-[l-y-(l-2/)-(y-l)]i. 

25.  .5  a  —  J  a  +  .3  a6  —  .3  a  —  [.25  a  4-  c  -f  1.5  a]  I . 

99.  Introduction  of  Parentheses. 

The  value  of  a  polynomial  is  not  changed  : 

1.  If  any  number  of  terms  with  their  signs  unchanged  are 
grouped  in  a  parenthesis  preceded  by  the  sign  +. 

2.  If  any  number  of  terms  with  their  signs  changed  are 
grouped  in  a  parenthesis  preceded  by  the  sign  — .     Sec.  97. 

For  example : 

2a+    5?)-6c  =  2a+(56 -6c). 
2a+    5&-6ci=2a-(-56  +  6c). 
4a-7m  +  2x  =  4a-(7m-2x). 

100.  The  fact  that  the  terms  of  a  polynomial  may  be  grouped 
as  stated  in  Sec.  99,  without  changing  the  value  of  the  poly- 
nomial, is  called  the  Associative  Law  of  Addition. 


WRITTEN    EXERCISES 

Write  as  a  plus  a  parenthesis : 

1.  a  +  6  6-4c.     3.    a-46-c+5cZ.     5.    a4-9  6-7c+3(?. 

2.  a  — 36  +  7 c.     4.    a-86-hc  — 1.       6.    a4-26  +  7cH-4. 

7-12.    Write  each  of  the  expressions  in  Exercises  1-6  as 
a  minus  a  parenthesis. 

13-18.  In  each  of  the  expressions  in  Exercises  1-6  place  the 
terms  involving  b  and  c  in  a  parenthesis  preceded  by  the 
sign  -. 


SUBTRACTION  61 

Group  the  like  terms  in  a  parenthesis  preceded  by  a  minus  sign : 

19.  bc  —  a'^-\'d-^3a'^  +  c  —  5  a\ 

20.  x^  —  2  xy -^  y"^  —  3  xy -\- z. 

21.  ab-\-bc  —  2ab  —  cd-{-5  ab. 

22.  Collect  in  a  parenthesis  the  like  terms  mxiny^—2x-i-x'^ 
—px-\-z^  —2jqx. 

23.  Collect  the  like  terms  in  x^,  also  those  in  y,  each  in  a 
parenthesis  preceded  by  a  minus  sign : 

x^  —  2/^  —px^  -^  2  y  -\- 5  x"^  —  xy  -\-  py, 

24.  Taking  the   terms  of   this  expression  in   order,  group 
each  pair  in  a  parenthesis : 

3  a^  —  b  —  c^  —  cP-\-bG-\-ac  —  x^  —  xyz. 

101.    The  sum  or  the  difference  of  terms  alike  with  respect 
to  certain  letters  may  be  indicated  by  the  use  of  parentheses. 

EXAMPLES 

1.  Add  ax  and  bx. 

Addend  ax 

Addend hx 

Sum       {a  +  h)x 

2.  Subtract  (c  +  n)  xy  from  (6  +  2  c)  xy. 

(6  +  2  c)xy 

(c  -f    n)xy 

(b  +  c—  n)xy 

WRITTEN   EXERCISES 
Add: 

1.  {a-\-b)x  3.    (a  +  c)  xy  5.    (a  +  c)  (m^  +  n) 

bx  (a  —  c)  xy  c  (m^  -f  n) 

2.  (l-{-2b)x^      4.    (a -{- b)  (p -\- q)      6.         (yn  -  14. -\- p)  x'^y 

—  bx"^  (c  +  d)  (p  H--  q)  {  —  m  -\-  2  n  —  p)  x^y 

7.    Collect   the  terms  in  x  and  put  the  coefficients  in  a 
parenthesis  with  the  proper  sign  : 

a  —  x-\-bx  —  abx. 


62  A  HIGH  SCHOOL  ALGEBRA 

8.  Similarly,  collect  the  terms  in  x,  also  those  in  y : 

ax  —  by  —  bx—  ay. 

9.  Similarly,  collect  the  like  terms  of  each  kind : 

3a-2b-6a-c-5b-\-10c. 

10.    Similarly,  collect  the  like  terms  in  x^,  and  also  in  xy: 

2x^—  xy  +  ax^  —  3xy  —  bx-{-2  axy. 
Group  in  parentheses  the  same  powers  of  x  in  each : 

11.  ax^  —  bx^  -\-  ex  —  ax^  +  bx^  —  dx. 

12.  x^  —  3x^-\-ax^—2x  +  cx^  +  bx. 

13.  mx^  +  2px  -\-  qx^  —  qx  —pa^  —  px". 

14.  a^  —  a)^  +  bx^  —  ax'^  +  bx^  —  px. 

Collect  in  alphabetical  order  the  coefficients  of  x,  also  of  y : 

15.  2ax  —  by-{-3bx—4:ay  —  cx-\-2cy. 

16.  mx -\- py  —  qx -{- ny  —  5  px -\- 4:  nx. 

Subtract: 

17.  {a  +  2b)xP       19.      {a  +  c){m^  +  n)   21.    {a-^c){x-{-y). 

bo^  (2a  +  c)(m2  +  n)  2c{x  +  y) 

18.  {2m-4:)x'^y    20.    (a+    c)^  22.       {l-\-b     -{-c)pV 

(37 -7i)x'y  {a  +  2c)xP  (g  _  2  6  -  2)pV 

First  collect  like  terms,  then  subtract : 

23.  ax  +  bx  from  ex  +  dx. 

24.  aa?  —  bx  from  ca?  —  dx. 

25.  «aj  —  3  6a;  from  ax  — 4:  bx. 

26.  aa;  4-  4  6a;  from  2  aa;  —  6a;  +  ex. 

27.  aa;  H-  2  2/  —  3  a;  from  ax  —  y-{-2x. 

28.  From  (a  +  6)a;^  plus  (c  —  a)a;2  subtract  (6  —  c)af. 

29.  From  (a  —  2  6)  xy  plus  (6  —  2  a)  a;?/  subtract  (a  +  6  -f-  c)  a;?/. 

30.  From  (a -{- b)  x -\- (c -{- d)  y  plus  2bx  —  2cy  take  2  aa; 

-(2d-c)2/. 


SUBTRACTION  63 

REVIEW 

ORAL  EXERCISES 

Subtract : 

1.  5n  4.   a; +  2/  7.    11  a +  10  &     10.   a  — x 
16  n                       x  —  x  a-\-   Sb  a  —  1 

2.  7  a  5.       a- 7  8.     Sa-b         11.  p^ -  q^ 
-9a                    2a-3  Sa  +  b  f  +  q^ 


8. 

3a  — 6 

3a  +  6 

9. 

ex-  dy 

X—    y 

3.   7  a—  5  b         6.      x^  -hy         9.      ex  —  dy        12.    ax—b 
a  — 2  b  2x^  —  y  ^—    y  bx  —  c 

13.    {a-2b)xy      14.    (2  a  +     6-c)a^     15.    (a  +  2  +  2c)/* 
(5-    q)a;y  (a-2  6-c)a;^  (l-ha)y^' 

WRITTEN    EXERCISES 

* 

Kemove  the  parentheses : 

1.  (a  +  6_c)  +  («-6  +  c).  5.  2y-{ltx-{-y). 

2.  (a-\-b-c)-{a-b-hc).  6.  {^tx -iy)-(iy  -  ^tx). 

3.  7a  — 36— (5a +  36).  7.  m  — [(a  — 6)  — (c  — ?n)]. 

4.  3«-7-(9aj-ll).  8.  m +  [(a  -  6)  +  (6  +  d)]. 

9.    6a;  +  52/-32-(5a;-32/  +  22). 
10.   6a2-(3a6+2ac)-(2ac  +  3a6). 
Subtract  and  test : 

11.  3x'  +  2xy-i-z^  13.    6c  +  3c?  +  a-30 

a;^-2a;y  3c-2d-2a 

12.  m  — 3n+p  — 7  14.        4:X-{- Sy  —  Az-{-S 
m  — 4n— p  +  8  — 7a;  —  3y  —  2;g  +  17 

15.  8aj'-9a?2+    o^^-a;  +16  +  a^ 

2x  -7     +5a;^-a^-a^  +6a^ 

16.  5.65 a  +     7|6-  27fc+  .76a;-       l^y 

4^  g  -  9.38  b  +  2.65  c  -  131  a;  -  0.375  y 


64  A   HIGH   SCHOOL   ALGEBRA 

17.  G-roup  the  last  four  terms  in  a  bracket  and  the  last 
three  in  a  parenthesis  : 

mp  —  3  mn  -{- 2  pq -{- m^  —  n^. 

18.  Group  as  the  difference  of  two  trinomials  without  chang- 
ing the  order  of  the  terms : 

QC?  -\-  2  xy  -\-  y^  —  x^  -\-  2  xy  —  y"^. 
Test  by  removing  the  parentheses  and  point  out  any  errors  : 

19.  a^— a^  +  a  — l=(a''^  — 1)  — (a^  +  a). 

20.  a2-&2_|_26c-c2  =  a2_(52^25c-c2). 

21.  From  {2h-\-c)x  take  {h  —  c)x.  ■ 

22.  From  (a  +  2  5  —  c)  xy  take  (2  a  —  5  +  2  c)  xy. 

23.  From  the  sum  of  (2  a  +  h)  x^  and  (c  —  h)  x^  take 

(a  4-  &  —  c)x^. 

24.  Collect  into  the  first  member  the  terms  in  x  and  unite : 

15-{2a-i-b)x-^3cx  =  20-^{c-2b)x-4:ax. 


SUMMARY 

The  following  questions  summarize  the  definitions  and  pro- 
cesses in  this  chapter: 

1.  Define  subtraction;  also  differeiice.  Sees.  91,  92. 

2.  Explain  how  to  subtract  monomials.  Sec.  93. 

3.  Explain  how  to  subtract  polynomials.  Sec.  95. 

4.  Explain  the  test  of  subtraction  by  arbitrary  values. 

Sec.  96. 

5.  State  the  sign  rules  for  removing  parentheses. 

Sec.  97. 

6.  State  the  sign  rules  for  introducing  parentheses. 

Sec.  99. 


CHAPTER   VII 
EQUATIONS 

102.  Practice  in  writing  expressions  containing  positive 
and  negative  numbers  is  a  necessary  preparation  for  solving 
equations  involving  these  numbers. 

EXAMPLE 
By  how  much  does  6  +  3  exceed  6  —  3  ? 
Indicating  the  difference  and  simplifying  : 

&  +  3-(&- 3)  =6  +  3-6  +  3  =  6-6  +  3  +  3  =6. 

WRITTEN   EXERCISES 

By  how  much  does 

1.  2  a;  — 5  exceed  a;— 5?  3.   2  a;  —  5  exceed  a;  — 6? 

2.  2  a  — x  exceed  2  a  4- a;?  4.   4  a  — 1  exceed  2  a  — 20? 

5.  B's  age  is  3  a;  —  1  years.      What  was  it  5  years  ago  ? 
What  will  it  be  x-j-2  years  hence  ? 

If  A  is  ?i  +-  2  years  old  and  B  is  2  n  —  1  years  old,  express  : 

6.  A's  age  4  years  ago.  8.    A's  age  5  years  hence. 

7.  B's  age  4  years  ago.  9.   B's  age  5  years  hence. 

10.  The  sum  of  their  ages  now. 

11.  The  sum  of  their  ages  4  years  ago. 

12.  The  sum  of  their  ages  5  years  hence. 

13.  How  much  older  is  B  than  A  ? 

If  A  has  d-\-25  dollars  and  B  has  2d  — 6  dollars,  write  the 
equation  expressing  that : 

14.  A  and  B  together  have  $  100. 

15.  A  has  $20  more  than  B. 

16.  B  has  $  20  less  than  A. 

65 


66  A   HIGH   SCHOOL   ALGEBRA 

17.  B's   money   increased  by    $15   equals   A's   money  de- 
creased by  $10. 

18.  If  A  loses  $10  and  B  gains  $10,  they  will  have  the 
same  amounts.  ^ 

19.  Express  the  next  larger  integer  after  n.     Also  the  sec- 
ond one.     Also  the  third. 

20.  Express  the   next  integer  smaller   than  ?i;  the  second 
integer  smaller  than  n;  the  third. 

Express : 

21.  The  sum  of  two  consecutive  integers  is  27. 

22.  The  sum  of  three  consecutive  integers  is  15. 

23.  The  next  larger  integer  after  ti  —  5 ;  after  n  —  1. 

24.  The  next  integer  smaller  than  n  —  5 ;  than  n  —  1. 

25.  The  sum  of  n  —  4  and  the  two  next  smaller  integers. 

103.   Certain  classes  of  numbers  are  represented   by  alge- 
braic formulas. 

EXAMPLES 

1.  2  n  represents  all  even  numbers,  if  n  be  given  all  positive  integral 
values. 

2.  2  n  —  1  represents  all  odd  numbers,  if  n  be  given  all  positive  integral 
values. 

3.  n  +  —  +  -^  represents  any  number  with  two  decimal  places,  where 
n  is  the  whole  number  and  m  and  p  are  the  decimal  figures. 

4.  —  represents  any  fraction.    If  n  and  m  are  integers  and  n  <  m,  it 
m 

is  a  proper  fraction. 

5.  n2  represents  square  numbers.     It  represents  the  squares  of  all  in- 
tegers if  n  has  all  positive  or  all  negative  integral  values. 


WRITTEN   EXERCISES 

1.  In  2  n  substitute  for  n  the  numbers  that  will  produce 
the  even  numbers  between  99  and  111. 

2.  In  2  71  substitute  for  n  the  numbers  that  will  produce 
the  multiples  of  4  between  15  and  25. 


EQUATIONS  67 

3.  If  the  negative  integers  be  substituted  for  n  in  2n, 
what  kind  of  numbers  are  produced? 

4.  When  n  is  1,  what  is  the  value  of  2  n  + 1  ?  When  n  is 
2 ?  When  riisS?  4?  0?  What  class  o-f  numbers  does 
2/1  +  1  represent  when  ^  may  be  zero  or  any  positive  integer  ? 

5.  Write  a  formula  to  represent  all  positive  cube  numbers. 
Can  the  values  of  the  letter  be  negative  ?  If  they  are  taken 
as  negative,  what  class  of  numbers  is  produced  ? 

6.  Can  you  substitute  in  n^  any  numbers  that  will  make 
the  series,  —  1,  —  4,  —  9,  —  16,  and  so  on  ? 

7.  What   number   does  ''^ -^Tk-\- :^^  become  when  7i  =  l, 

m  =  2,  p  =  3?     Also  when  w  =  63,  m  =  0,  p  =  S?     Also  when 
n  =  0,  711  =  2, p  =  5? 

8.  Write  a  formula  to  represent  any  number  with  three 
decimal  places. 

9.  If  q  is  10  in  ^,  what  is  the  greatest  integral  value  p  can 

Q 
have  and  make  -^  a  proper  fraction  ? 

10.   AVhat  kind  of  a  fraction  is for  all  positive  integral 

1  r      o  n-\-l 

values  01  7i  c 

104.   The  processes  with  negative  numbers  may  be  applied 
in  substitution. 

EXAMPLES 

1.  What  is  the  value  of  d  in  d  =  vt,  when  v  =  —  S  and  ^  =  8  ? 

Given  d  =  Vt.  (1) 

When  t?  =  -  8  and  <  =  8,  fZ  =  —  3  •  8.  (2) 

Therefore,  d  =  -  24.      (SeC.  75.)  (3) 

2.  What  is  the  value  of  ^  in  d  =  vtj  when  d  =  — 16,  v=—2? 

Given  d  =  Vt.  (1) 

Dividing  hy  v,  t  =  -  -  (2) 

V 

1 A 

When  rf  =  -  16  and  t'  =  -  2,     t  =  ^—-  •  (5) 

—  2 

Therefore,  t  =  8.      (Sec.  77.)  '  (^) 


68  A   HIGH   SCHOOL  ALGEBRA 

3.   Eind  a  in  a  =  ttt^,  when  r  =  —  2. 

Whenr-  =  -2,  ^  =  7r(— 2)"^.  (i) 

Therefore,  a  =  3. 1416  x  (-  2)  (-  2).     (Sec.  75.)    {2) 

=  12.5664. 


ORAL  EXERCISES 

1.  In  IV  =fs,  find  w  when  /=  6  and  s  =  —  3.  Also  find  w 
when  /=  12  and  s  =  —  ^. 

2.  In  w  =fs,  find  /  when  iv  =  20  and  s  =  —  5. 

3.  In  d  =  v^,  find  d  when  v  =  —  12  and  ^  =  J.  Also  find  t 
when  d  =  —  2  and  v  =  —  ^. 

4.  In  s  =  i  a6,  find  s  when  a  =  6  and  6  =  —  5.  Also  find  s 
when  a  =  —  8  and  &  =  —  3. 

5.  In  x  =  2  ay,  find  a;  when  ?/  =  — 15.  Also  find  y  when 
a;  =  -10. 

6.  In  3x=az,  find  a;  when  z=—9.     Also  find  2;  when  a;=4a. 

WRITTEN    EXERCISES 

1.  In  v  =  Trr^a,  find  v  when  v  =  3.1416  and  ?-  ==  —  25,. 

2.  In  V  =  Trr^a,  find  a  when  'y  =  9.4248  and  r  =  —  h 

3.  In  A:  =  1^  mv"^,  find  A;  when  m  =  —  12  and  v  =  3.  Also 
find  k  when  m  =  —  15  and  y  =  —  2.  Also  find  A;  when  m  =  20 
and  'y  =  —  5. 

4.  In  a  =  Tr?*^,  find  a  when  r  =  —  4 ;  also  when  r  =  —  ^. 

5.  In  c  =  2  ttv,  find  c  when  ?•  =  20  ft. ;  also  when  r=  —15  ft. 

6.  In  s  =  ^  ah,  find  s  when  a  =  — 17  and  6  =  —  12 ;  also 
when  a  =  15  and  6  =  —  9. 

7.  In  s  =  i  ah,  find  a  when  s  =  —  42  and  h  =  —  T. 

8.  In  ic  =  3  2/2,  find  a?  when  ?/  =  14 ;  also  when  2/  =  —  14. 

9.  In  3 a;  4-1  =4?/— 4,  find  a;  when  2/ =  —  5. 

10.  In  ax=:hy-\-hc  +  a,  find  x  when  y  =  —  c. 

Q 

11.  In  ay  —  h  =  hx  +  c  find  a;  when  y  =  --     Also  find  2/  when 

a 

aj=-l. 


dx           =11+4  =  15, 
X           =6. 

(Sec.  67.)  (2) 
(5) 

5x  +  21=    6 

(^) 

21  =21 

EQUATIONS  69 

105.  The  processes  with  negative  numbers  may  be  applied 
in  solving  equations. 

EXAMPLES 

1.  Solve  for  a;:  3a;-   4  =  11  (i)' 

Add  4  to  both  members,  4=4 

Then, 

and  c. 

Test.     3  •  5  -  4  =  11. 

2.  Solve  for  x: 

Subtract    21     from     both 
members. 

Then,  bx  =    6-21=- 15,   (Sec.  71.)   (3) 

and  X  =-3.  (Sec.  77.)   (3) 

Test.     5  (-3)+ 21  =6. 

The  effect  of  adding  or  subtracting  as  above  is  equivalent  to  taking  a 
number  from  one  side  of  the  equation  and  placing  it  on  the  other  with  the 
sign  changed,  and  the  term  transpose  is  often  used  to  name  this  process ; 
compare  steps  (1)  and  (2)  in  each  example.  But  it  is  much  better  at 
first  in  explaining  the  work  to  say  that  a  certain  number  has  been  added 
to,  or  subtracted  from,  both  members  of  the  equation. 

3.  Solve  for?/:  2y-6  =  3y-{-l.  (1) 

Subtracting  3  y  from  both  members,         _  y  _  6  =  1.  (Sec.  71.)  (^) 

Adding  6  to  both  members,  —y  =  7.  (See.  67.)  (3) 

Multiplying  both  members  by  -1,  y  =—7.     (Sec.  75.)  (4) 

Test.  2  (- 7)  -  6  =  3(*- 7)  +  1. 

The  change  from  step  (3)  to  step  (4)  follows  directly  from  the  meaning 
of  relative  numbers.  That  is,  if  negative  y  equals  positive  7,  then  positive 
y  must  equal  negative  7.  But  since  the  sign  of  a  number  is  changed  by 
multiplying  or  dividing  it  by  —  1,  it  is  customary  to  refer  such  a  change 
of  signs  to  one  or  the  other  of  these  processes. 

ORAL  EXERCISES 

Solve  : 

1.  a:-fl  =  -8.  6.  7  a;  4-5  =  - 9.     11.  x  — a  ^b  —  c. 

2.  aj-2  =  -5.  7.    2-3x  =  -4:.    12.  ax-2b=c~Sb. 

3.  2a; -2  =  6.  8.  5 -4a;  =  21.       13.  4  c-2  6a;=8-4  6a;. 

4.  2y-l  =  ll.  9.  |2/-4  =  8.         1^.  y-a=-b. 
6,  3  2/- 5  =  13.  10.  5^  +  5  =  -20.  15.  ay +  6  =  - c. 


70  A  HIGH   SCHOOL  ALGEBRA 

WRITTEN    EXERCISES 
Solve  for  x : 

1.  2  a;  — 1=3  a; -7.  6.    a;  +  7  =  2  — jaj. 

2.  5a;  +  3  =  2a;-15.  7.   |-3a;  =  4a;-i 

3.  4  a;  — 5  =  10 -a;.  8.   fa; -8  =  —  17  — |  a;. 

4.  i a;  +  6  =  a;  —  18.  9.ax—b  =  ^ax-{-c. 

5.  f  a;  —  5  =  20  —  a;.  10.    ax  -\-m  =  hc—  (mx  +  m). 
Solve  for  the  letter  in  each  case : 

11.  a;-3  +  4a;  +  21-2a;  =  0.  17.  8-2p  +  6-5p  =  2L 

12.  2/  +  8  +  62/-4-32/  =  0.  18.  7  s  -  16  =  3 -3  s- 29. 

13.  2  2/- 5-2/  +  20  =  -15.  19.  35 -4  2/  =  6?/ +  5  +  5?/. 

14.  m  — 7  +  3m  +  42  =  — 4  m.  20.  2 +  :«  —  5  =  8  a;+ 11. 

15.  5^  +  6-3^-10  =  ^.  21.  10  +  ^-4  =  -52;-12. 

16.  12-7i  +  8-4/i  =  -10.  22.  S-Sw  =  2S-2Sio. 

106.    The  process  of  removing  parentheses  appUes  to  equa- 
tions. 

EXAMPLE 

Solve  :  a;  -  (2  a;  +  6)  =  3  -  (4  a;  -  6).  (1) 

Eemoving  the  parentheses,    x  —  2aj  —  6  =  3  — 4a;  +  6.    (Sec.  97.)        (^) 
Then,  .  S  X  =  16, 

and  X  =  5. 

Test.  5  -  (2  •  5  +  6)  =-  11  =  3-  (20  -  6). 

WRITTEN   EXERCISES 

Solve  and  test : 

1.  2  a;  -  (2  +  a;)  =  6.  2.  2y  -  (y -\-6)  =2  y -7. 

2.  ia;  +  (2-a;)  =  6.  10.  3  a;-(2  a;+ 8)  =  2 -4  a;. 

3.  4a;-(2a;  +  l)  =  5.  11.  7 +  t  =  3- {2t-17). 

4.  5  -  (4  a;  +  2)  =  4.  12.  5  -  3  s  =  6  -  (s  +  15). 

5.  7  -  (4  .T  +  3)  =  0.  13.  ;^  -  (6  -  5  2  -  18)  =  30.      - 

6.  6  2/ +  (4  2/ -10)  =  10.  14.  4p-(3p  +  l)  =  2p  +  l. 

7.  12  2!  -  (2  +  6  ^)  =  16.  15.  5  ^o  -  (w  +  6)  =  2  -  (w;  +  1). 

8.  2J  +  (2i-50  =  0.  16.  z-6  =  z-(Sz  +  17). 


EQUATIONS  71 

17.  -(x-\-5)==S  +  (x-5).   Id.  2y-(y  +  5)  =  'i-(y-S). 

18.  15- (2 +  12)  =  42 -20.   20.  16  -  (2  a;-h8)=3  a;4-(4  a;-l). 

107.   In  solving  problems  by  means  of  equations,  the  paren- 
thesis is  often  used. 

EXAMPLE 

The  sum  of  two  numbers  is  100,  and  3  times  one  of  them  is 
7  times  the  other  ;  what  are  the  numbers  ? 

Solution.     1.    Let  x  be  the  smaller  number. 

2.  Then,  100  —  a:  is  the  larger  number. 

3.  Then,  7  a;  is  7  times  the  smaller  one  and 
3(100  —  aj)  is  3  times  the  larger  one. 

4.  .-.  7  X  =  3(100  —  x),  according  to  the  problem. 
6.   .■.lx  =  SOO-Sx.     (Sec.  24.) 

6.  .-.  10x  =  300. 

7.  .-.  a;  =  30,  and  the  numbers  are  30,  70. 

'      Test.     30  +  70  =  100,  and  7  x  30  =  3  x  70.     Therefore  the  numbers 
found  fulfill  the  conditions  of  the  problem. 

The  use  of  the  parenthesis  is  seen  in  the  third  and  fourth  steps. 

WRITTEN    EXERCISES 

1.  Write  an  equation  stating  that  if  the  cost  (c)  of  a  lot  be 
diminished  by  $200  and  the  remainder  multiplied  by  5,  the 
result  will  be  the  value  (v)  of  the  house. 

2.  Write  the  equation  which  states  that  c  times  the  sum  of 
a;  and  a  equals  d. 

3.  A  father  is  twice  as  old  as  his  son ;  11  years  ago  he  was 
three  times  as  old  as  his  son.     Find  the  age  of  each. 

4.  If  a  certain  number  is  diminished  by  9  and  the  remainder 
multiplied  by  9,  the  result  is  the  same  as  if  the  number  were 
diminished  by  6,  and  the  remainder  multiplied  by  6.  Find 
the  number. 

5.  Divide  48  into  two  such  parts  that  one  part  shall  exceed 
the  other  by  6. 

6.  A  man  is  four  times  as  old  as  his  son ;  in  18  years  he 
will  be  only  twice  as  old.     Find  the  age  of  each. 

6 


72  A   HIGH   SCHOOL  ALGEBRA 

7.  Two  men  enter  a  partnership  and  together  furnish  a 
capital  of  $  5000 ;  twice  what  one  furnishes  is  3  times  what  the 
other  furnishes.     How  much  does  each  furnish  ? 

8.  The  sum  of  two  numbers  is  40  ;  the  smaller  number,  x, 
is  ^  of  the  larger  number.  Write  the  equation  needed  to  find 
X.     Find  the  numbers. 

9.  A  rectangular  lot  is  20  ft.  longer  than  it  is  wide.  Using 
X  to  represent  the  width,  state  what  represents  the  length. 
Write  an  equation  stating  that  4  times  the  width  equals  2  times 
the  length.     Find  the  dimensions  of  the  lot. 

10.  In  a  certain  post  office  there  are  three  rates  of  pay :  $  50, 
$  100,  and  $  150  per  month.  There  are  5  more  men  receiving 
$100  than  $150,  and  2  more  receiving  $50  than  $100;  the 
monthly  pay  roll  is  $  1150.  Letting  x  represent  the  number 
receiving  $  150,  write  the  equation  needed  to  find  x. 

108.  The  solution  of  problems  must  be  proved  by  substitut- 
ing the  results  in  the  conditions  of  the  problems. 

EXAMPLE 

1.  A  commission  merchant  remitted  $475  as  the  proceeds 
of  a  sale  of  500  bu.  of  potatoes  after  deducting  his  commission 
of  5%.     For  how  much  did  he  sell  the  potatoes  ? 

Solution.     1.   Let  x  =  the  number  of  dollars  received  for  the  potatoes. 

2.  Then,       x—  .06x  =  the  amount  remitted. 

3.  .'.  X  —  .06x  =  475,  according  to  the  problem. 

4.  .-.  .95  X  ='475. 

5.  Dividing  by  .95,  x  =  500. 

6.  .*.  the  potatoes  sold  for  $  500. 
Test.     500  -  .05  •  500  =  475. 

This  merely  tests  the  correctness  of  the  work  after  step  3 ; 
it  does  not  test  the  correctness  of  the  equation  in  step  3 ;  to 
do  this  the  result,  500,  must  be  tested  in  the  conditions  of  the 
'problem  itself. 

Thus,  (1)  5%  of  1500  =  $25,  (2)  !$  500  -  1 25  =  $  475,  the  proceeds. 
If  equation  3  had  been  incorrectly  written,  the  result  found  in  step  6 
might  have  been  a  correct  solution  of  equation  3  without  giving  the  correct 
proceeds. 


EQUATIONS  73 

WRITTEN    EXERCISES 

Solve  and  test : 

1.  If  A  has  a;  dollars  and  B  has  twice  as  much,  express  what 
the}^  both  have.  If  this  amount  is  $120,  write  and  solve  an 
equation,  thus  finding  what  each  has. 

2.  The  sum  of  a  number  and  .05  of  itself  is  210,  what  is  the 
number  ? 

\  3.    Separate  the  number  72  into  two  such  parts  that  one  part 
shall  be  ^  of  the  other. 

4.  Divide  100  into  two  parts  such  that  one  is  24  less  than 
the  other. 

V  5.   Three  times  a  number  less  twice  the  difference  between 
the  number  and  5  is  30.     Find  the  number. 

6.  Find  a  number  such  that  when  9  is  added  to  three  times 
the  number  the  sum  is  42. 

7.  A  tree  120  ft.  high  was  broken  so  that  the  length  of  the 
part  broken  off  was  four  times  the  length  of  the  part  left  stand- 
ing.    Find  the  length  of  each  part. 

'^"^^  8.  Three  men  together  have  $  1800.  The  first  has  twice  as 
much  as  the  second,  and  the  third  has  as  much  as  the  first  and 
second.     How  much  has  each  ? 

9.  A  man  was  hired  for  50  da.  Each  day  he  worked  he 
was  to  receive  $2  and  each  day  he  was  idle  he  was  to  forfeit 
50 j^  instead  of  getting  his  wages.  At  the  end  of  the  50  da.  his 
balance  was  $  80.     How  many  days  did  he  work  ? 

10.  A  farmer  bought  16  sheep.  If  he  had  bought  4  sheep 
more  for  the  same  money,  each  sheep  would  have  cost  him  one 
dollar  less.     How  much  did  he  pay  for  a  sheep  ? 

11.  A,  B,  and  C  bought  a  summer  cottage  for  $3000.  B  pays 
twice  as  much  as  A,  and  C  pays  as  much  as  A  and  B  together. 
How  much  does  each  pay  ? 

12.  A  tree  90  ft.  high  was  broken  so  that  the  length  of  the 
part  broken  oif  was  5  times  the  length  of  the  part  left  standing. 
Find  the  length  of  each  part. 


74  A  HIGH   SCHOOL   ALGEBRA 

HISTORICAL  NOTE 

The  earliest  attempts  to  solve  equations  mark  the  beginning  of  algebra. 
About  1700  B.C.  there  lived  an  Egyptian  priest  named  Ahmes.  He  was 
probably  a  teacher  of  mathematics,  and  was  the  first  writer  on  the  subject 
whose  work  is  still  in  existence,  for  there  has  been  recovered  from  the 
ancient  pyramids  a  manual  written  by  him  which  contains,  among  other 
mathematical  subjects,  a  simple  treatment  of  equations.  Ahmes'  work, 
called  Directions  for  obtaining  the  Knowledge  of  all  Dark  Things^  is 
written  in  hieroglyphics  on  papyrus,  and  is  now,  thirty -six  centuries  after 
its  production,  the  property  of  the  British  Museum.  It  was  deciphered  by 
the  German  scholar,  Eisenlohr,  in  1877. 

The  equations  of  Ahmes  were  expressed  in  words,  not  in  symbols  ;  for 
example,  one  of  his  problems,  when  translated,  reads  :  "  Hau  (literally 
heap^  meaning  unknown  quantity)  its  seventh,  its  whole,  makes  nineteen," 

which  in  present-day  algebraic  notation  is  x  ■{■-  —  19.     There  are  few 

rules  in  Ahmes'  handbook  to  show  how  his  results  were  obtained,  but  it  con- 
tains certain  tables  of  numbers  that  indicate  the  processes.  For  example, 
he  could  write  only  unit-fractions,  like  |,  ^,  and  ^,  other  fractions  being 
expressed  by  the  sum  of  unit-fractions.  Accordingly,  he  gives  the  solution 
of  the  above  equation  as  follows  : 

(1)    8    \x  =  \^  (2)    }.x  =  2    \    \  (3)   a;  =  16    I    i 

(In  the  first  step  the  adjacent  numbers  are  supposed  to  be  multiplied, 
and  in  the  other  steps  they  are  to  be  added. ) 

Ahmes  was  greatly  hindered  by  lack  of  suitable  notation  and  other  lim- 
itations, and  knowing  nothing  of  negative  number,  he  could  not  solve  so 
simple  an  equation  as  aj  -1-  3  =  1. 

Little  progress  was  made  in  solving  equations  during  the  twenty  centu- 
ries from  Ahmes  to  Diophantos.  During  the  Middle  Ages  the  Arabs  and 
Hindoos  simplified  the  solutions  of  Diophantos,  but  the  perfection  of 
modern  symbols  and  the  explanation  of  positive  and  negative  roots  re- 
mained for  the  mathematicians  of  the  sixteenth  century. 


CHAPTER   VIII 
MULTIPLICATION 

MULTIPLICATION  OF  MONOMIALS 

109.  Preparatoky. 

1.  5.3^=()^.       3.   5-3ft.=()ft.       5.    5.3/=()/. 

2.  6'3d={)d.      ^.    7  '5x=()x,  6.   6'6abc={  )abc. 
7.   3a-46  =  3.4a6  =  ?                8.   la- 86  =  4- of  8  a6  =  ? 

9.   4:X  ■  6y  '  ^z  =  4:  '  6  '  ^ xyz  =  (  ) xyz. 

110.  To  find  the  product  of  two  monomials,  take  the  product 
of  their  literal  parts  for  the  literal  part  of  the  product,  and  the 
product  of  their  coefficients  for  the  coefficient  of  the  product. 

Prefix  to  the  result  the  sign  +  or  —  according  to  Sec.  75. 

111.  Repeated  Factors.  If  the  same  letter  occurs  more  than 
once  as  a  factor  in  the  product,  it  should  be  written  only  once, 
with  the  proper  exponent  (Sec.  14). 

For  example : 

-3a  .  2a&  z=-Q  aah  =-Q  a'^h. 
a%^ '  ab^  =  aabbb  •  abb  =  aaa  •  bhbbb  =  a^b^ 

ORAL  EXERCISES 

State  the  products : 

1.  f'-t'.  7.  4:a'2ax.  13.  (4p2)(8p4). 

2.  2/5 . 2/2.  8.  4a^  .  —  2  x\  14.  {am%—  arn?). 

3.  x^y  '  y^.  9.  a^x  •  —  a^x^.  15.  2aV  •  ^a'^ir'. 

4.  8  c.  8  c.  10.  12  ab' 2  ab\  16.  1  a^  -  -  a\ 

5.  la-^ab.  11.  3xy^  '  —Bx'^y.  17.  15v^'v\ 

6.  a^  •  —  a\  12.    2  a^  •  a\  18.    13  x'  -  a?. 

75 


76  A   HIGH   SCHOOL   ALGEBRA 

WRITTEN   EXERCISES 

Multiply  as  indicated : 


1. 

48a63.  -  10  bc\ 

4. 

16  a''- 15  a'c. 

7. 

30  .r2. 

—  4aa^. 

2. 

24ta^bC'12c. 

5. 

36x'''Sxz\ 

8. 

65  a"^^ 

-  4  a'b. 

3. 

214  6c.  6 ftV. 

6. 

12  m.^ '  -  16  m^^. 

9. 

27  a'' 

-63a'b\ 

112.  Law  of  Exponents  in  Multiplication.  The  exponent  of 
any  letter  in  a  product  is  the  sum  of  the  exponents  of  that  letter  in 
all  of  the  factors. 

In  symbols  this  law  is  expressed : 

a"" '  a''  =  a'"■'"^ 

EXAMPLES 

1.   a^-a"^  =  o5+7  =  0^12. 

3.   a"  •  —  a'^x  •  axP-^  =—  a^  ■  a^  -  a^  -  x^  •  xp-^ 

=  _  a'^+^xP-^. 

113.  To  find  the  product  of  several  monomials: 

1.  Find  the  sign  of  the  product. 

The  sign  is  plus  when  the  number  of  negative  factors  is  even,  and 
minus  when  the  number  of  negative  factors  is  odd. 

2.  Fiyid  the  numerical  coefficient  of  the  product.    ■ 

3.  Find  the  literal  part  of  the  product. 

EXAMPLE 

Find  the  product  of  —2  a,  4-  3  6,  —2ab,  —  4  6c. 

1.  The  sign  is  — . 

2.  2  .  3  .  2  .  4  =  48. 

3.  a  •  b  '  ab  '  be  =  a%H. 

4.  .-.  -  2  a  .  3  6  .  -  2  a&  .  -  4  6c  =  -  48  aWc. 

114.  If  any  factor  is  zero,  the  product  is  zero. 
For  example  : 

3  •  0  =  0  ;  similarly,  —  a;  •  0  =  0,  and  x  -  y  -  z  •  0  =  0. 

This  follows  from  the  meaning  of  zero  Sec.  66.    For  if  3  —  3  =  0,  then 
(3  -  3)4  represents  0  •  4  ;  but  (3-3)4  =  12  -  12  =  0. 
Hence,  0  •  4  should  be  taken  to  mean  0. 


MULTIPLICATION  77 

ORAL  EXERCISES 

Multiply : 

1.  6,-3,5.,  5.  2x,  -Sy,  -2z.     9.    (-6),  (-6). 

2.  —7,3,  —1.  6.  ax,  —bx,  —ex.       10.    (— 4a^),  (— 4fl;). 

3.  a,  0,  -c.  7.   -a,  -a,0.  11.    (-2)3,  (-a/. 

4.  -6,-0,-4.  8.  (-g),(-$),(-Q).    12.    0^,  (-a))2,  (-0^)3. 

13.  ^,{—xy.  15.    2  ^ic,  4  ^ic,  —  graj^ 

14.  -x'',{-x)\  16.    m2,  -7i2^  (-p)2 

WRITTEN   EXERCISES 

Multiply : 

1.  (-  2)2 .  (-  3)3 .  (-  1)2.  6.    20  aj?/p^«  •  -  71  a;^?/'^^'-. 

2.  (-3a6)3.(-3a5)2.5a263.      7.  lb a^+^ • -12 a?^h  •  ^ a^h'^-K 

3.  a5a;2/  •  —  21  a^b^x'^y^  8.  17  a^^x^-^  -  —  5  aP-'^a?! 

4.  31  a^'af  •  —  15  cf-V-i.  9.   —  24  x'^y^-'^  -  —  8  a;'-^^/^' 

5.  —  4a2x''~^«  —  64a"~2x^         10.  12m2y^  •  — 15m?i^  •  —m^n", 

11.  —^x^y^z'  •  -f  ic^- V^^^-'  •  —  tV  ^y^' 

12.  121  sH^y  •  —  17  ^fz^  '  0. 

13.  Suppose  12  a^Wc  to  be  taken  as  the  product  of  4  ab"^  and 
3  a^c.     Test  this  |)roduct  by  letting  a,  b,  and  c  each  equal  1. 

14.  Also  test  it  by  letting  a  =  2,  6  =  3,  and  c  —  5.    Why  did 
not  the  test  applied  in  Exercise  13  reveal  the  error  ? 

115.    Signs  of  Factors.     It  follows  from  Sec.  75  that,  if  the 
product  of  two  factors  is  positive,  the  factors  must  have  like 
signs;  but  if  the  product  is  negative,  the  factors  must  have 
unlike  signs. 
For  example  : 

S  ax  =  2  a-  4:X,  or  (—2  a)(—4:x). 
-Ub^c=  (7  6)(-2&c),  or  (-7  6)(2  6c). 

WRITTEN   EXERCISES 

Write  a  set  of  two  factors  for  each  of  the  following : 

1.  bVi.  3.   Trr'h.  5.    15  00^2/.  7.    Uy^ 

2.  ^gf.  4.    |7r7^.  6.    -7mw2p.  8.    -48  05. 


78 

A  HIGH  SCHOOL   ALGEBRA 

9.    moi?. 

12.    \mv\          15.    l^alPc. 

18.    21a3a^. 

10.    -7rr\ 

13.    ^7rd\           16.    -8a36. 

19.    -mc?l 

11.   4  7rr2. 

14.    Q*ax^.           17.    14  6?-2. 

20.    2  7r?7i. 

MULTIPLICATION   OF  POLYNOMIALS 

116.  To  multiply  a  polynomial  by  a  monomial  multiply  each 
term  of  the  multiplicand  by  the  monomial  and  use  the  signs  ob- 
tained as  the  signs  of  the  j^roduct. 

For  example  : 

2a-Sb  +  5c 
6a 


12  a2  -  18  ab  +  30  ac 


117.  Distributive  Law.  The  fact  that  a  polynomial  is  mul- 
tiplied by  multiplying  each  of  its  terms  separately  and  taking 
the  algebraic  sura  of  the  partial  products  thus  found  is  called 
the  Distributive  Law  of  Multiplication. 

The  formula,  a(b-\-c)  =  ab  +  ac,  expresses  this  law  in  symbols. 

ORAL  EXERCISES 
Read  and  supply  the  numbers  for  the  blanks : 

1.  3a  +  46  4.   3a  +  46  7.    6a2  +  26 

_3 Sa  3  5 

()a  +  ()6 

2.  5a2  +  2  6 
_10 

3.  lOx  +  y 

5 

0^  +  ()y  ()^/  +  ()a;y  45()+5() 

118.  To  test  the  work  of  mtiltiplication,  use  arbitrary  values. 
The  product  of  the  values  of  the  multiplicand  and  the  multi- 
plier must  equal  the  value  of  the  product. 

Unity  is  the  easiest  number  to  substitute  for  the  letters  ;  it  tests  the 
coefficients  and  signs  in  the  work  of  multiplication.  It  does  not,  however, 
test  the  exponents  (see  Exercises  13  and  14,  p.  77)  ;  but  this  does  not 
impair  the  test  materially,  since  errors  in  exponents  alone  seldom  occur. 


{)a2  +  ()a6 

5.    10m  +  2w 

5  mn 

(  )  mhi  +  (  )  mn"^ 

6.    Aa^y  +  xy'^ 

Sxy 

18()H-6() 

8.    5?7i  +  3i)2 

3  mp 

(  )m'^p  +  (  )  mp^ 

9.   9x'-\-y^ 

5xy 

MULTIPLICATION  79 

WRITTEN   EXERCISES 

Multiply  and  test : 

1.  ax-\-4:                    4:.   7  ax +  3  a  7.   a2-f-2  6a^ 
3 5x 3  b'^x 

2.  a-{-b                       5.   8a2  +  262 
c ab 

3.  2a +  36  e.   7  ax  +  Sbx"*' 

4:C  cx 


8. 

3m2  +  r2 
4a 

9. 

vt 

10.  4.x\a-5x).       13.    2x(3x-{-5y).       16.    6^2(5^  +  18^3). 

11.  a  (2x3  +  1).  14.    (7  a; -5)2  a.  17.   -i-r(2  r^  +  ir). 

12.  a}h(o?c-bM).      15.    (4a;  +  502to.       18.    6  ^  (i^  _j_  i  ^^^^ 

119.  Special  Tests.  Certain  polynomials  have  properties 
which  aid  in  testing  the  work  of  multiplication. 

Thus,  in  {x  +  y)  (x^  —  xy  +  y'^)  =  x^  -{-  y^,  each  term  of  the  first  factor 
is  of  the  first  degree,  and  each  term  of  the  second  is  of  the  second  degree  ; 
hence,  if  each  term  of  the  product  were  not  of  the  third  degree,  the 
product  would  be  incorrect. 

120.  Expressions  all  of  whose  terms  are  of  the  same  degree 
are  called  homogeneous ;  when  factors  are  homogeneous,  their 
product  is  homogeneous,  and  its  degree  is  the  sum  of  the 
degrees  of  its  factors. 

ORAL  EXERCISES 

1.  Without  multiplying,  state  the  degree  of  each  term  in 
the  product  of  a  +  6  and  a^  +  6^. 

2.  Similarly  for  x^-{-xy  and  x-\-y.  Also  for  mn  +  n^  and  wi^+n^ 

3.  Without  multiplying,  determine  whether  or  not  ii(?-\-xy 
+  2/3  is  the  product  of  x^  -\-xy  +  if  and  x-\-y. 

'     WRITTEN   EXERCISES 

Multiply  and  test  first  by  seeing  whether  the  product  is 
homogeneous.  If  it  is  homogeneous,  test  further  by  substitut- 
ing arbitrary  values : 

1.    (a^  +  2  Z>3)  (a3  _  3  63).  2.    (x-^y)(x''- 3  xy +  5  y^). 


80  A  HIGH  SCHOOL   ALGEBRA 

3.  (Q(?-y'^){x^Jf.(Sxy-y'^).       5.    (a?  -5  ax-\-  x')(^  +  2  a?). 

4.  {ci-\-x)\a}-x').  6.    {x'-2xy-\-y''){x^-\-y^). 

121.  Removal  of  Parentheses.  If  a  parenthesis  used  to  indi- 
cate mult)  pi  (cat  1071  is  removed,  the  multiplication  must  beperformed. 

Thus:        7  a  -  5(9  a  -  4  6)  =  7  a  -  45  a  +  20  6  =-  38  a  +  20  &. 

WRITTEN    EXERCISES 

Remove  parentheses  and  unite  terms  as  much  as  possible : 

1.  3  +  5(6-4).  6.   4a- 12(7 -6a). 

2.  5  a:  +  3(11  £c  -  5).  7.   9(a  —  x)— a(5 -j- x), 

3.  7(4a-26)+106.  8.    -  5(2  a;  -  1)  +  3(4  a;  -  8). 

4.  6  //  -  7(4  ?y  +  3  t).  9.    a(b  +  c)—  c{a  +  6)-f  b{a  —  c). 

5.  11  -  3(7  -  2  a.^).  10.    2m-{4m-f  7(6m-l)J. 

11.  p|4  r  -  3  r(l  —  a)+  5  ar|. 

12.  a;2|;c2  _  2  a(2  a;  -  3  a)  j  -  a3(4  x  -  a). 

13.  _l0;a;-6[a;-(?/-^)]j+60f2/-(^  +  a;)S. 

122.  The  multiplication  of  literal  numbers  is  similar  to  the 
multiplication  of  numbers  expressed  by  figures. 

For  example  : 
Multiplication  with  FiGtrBES  Multiplicatioi^with  Letters  Test 

32  (S^a  +  2  6)Ox-»"H>0                                   5 

14  a  +  4:b                                                       5 

128=    4x32  Sa^+2ab            =a(Sa+2b) 

320  =  10x32  12a6  +  8&2.=  4&(3a  +  2&) 

448  =  14x32  3a2+14a&  +  862=  (a+46)(3a+2&)      25 

123.  To  multiply  by  a  polynomial,  multiply  by  each  term  of  the 
polynomial,  add  like  terms,  and  use  the  signs  obtained  as  the  signs 
of  the  result. 

Thus: 

2  a2  _|_  5  «  _  2 
gg  -  3  g  +  1 
2  a^  +  5  a3  _  2  a2 

-6a3- 15  a2+    6a 

2a2+    5a -2 


2  a*  -  a«  -  15  a^  +  11  a 


MULTIPLICATION"  81 


WRITTEN    EXERCISES 
Multiply  and  test : 


1. 

a  +  5 

10. 

X  -\-  a 

19. 

3  a6  +  4  62 

a-\-b 

x-\-b 

2  a6  -  3  &2 

2. 

a  +  b 

11. 

3a-{-x 

20. 

aj2  ^  3  a.  _  1 

a-b 

a-{-b 

.T  4-3 

3. 

a  —  b 

12. 

4a  +  5 

21. 

a;2  _  4  aj  4_  3 

a-b 

X  —  a 

ir  -2 

4. 

c  +  1 

13. 

m  +  3 

22. 

a;2  —  aa?  4-  6 

c-1 

3m  +  2 

a;  —  c 

6. 

x-\-2 

14. 

m^  —  n^ 

23. 

a;2  —  ace  4-  & 

x  +  2 

m^  -|_  n^ 

Sx-\-a 

6. 

z^^^ 

15. 

iC^  +  l 

24. 

f  +  tu  4-  u^ 

224-5 

^2-1 

t  -u 

7. 

2a-l 

16. 

2a  +  b 

25. 

a  +  b  —  c 

2a-l 

a-\-2b 

a  —  b  -\-  c 

8. 

12  + a; 

17. 

2a-     b 

26. 

a;2  4_  2/2  _  2;2 

12 -a: 

c-3a 

o;  4-?/  — 2! 

9. 

Sy-5 

18. 

Sx-\-2y 

27. 

ic?/  4-  2/2  4-  a;2 

2.V  +  4 

2x-^Sy 

x  —  y    4-2J 

124.  To  find  the  product  of  expressions  involving  literal  coeffi- 
cients and  exponents,  find  the  product  of  the  coefficients  and  add 
the  exponents  as  in  numerical  cases. 


EXAMPLES 

1. 

Multiply 

ax  and  (a 

-  b)x. 

(a  -  b)x 
ax 
a{a  -  6)x2 

2. 

Multiply 

a;"  4-  1  by 

x^-2. 

X-+1 
Xn_2 

x2n_  a;«  —  2 


82  A  HIGH   SCHOOL  ALGEBRA 

WRITTEN  EXERCISES 

Multiply : 

1.  (5  a  +  0?) (5  a  —  a;).  7.   z\z'  +  z% 

2.  {x  —  y){xr  —  ?/"*).  8.    {xc  —  X)cx. 

3.  {x^  -\-  y''){x  4-  y).  9.    (a  —  b)y  •  aby. 

4.  (4  X"*  +  3  2/") (4  a;'"  —  3  2/").      10.    (c  +  d)x  •  (c  —  d)x, 

5.  (aj'"  —  2/")(aj2"*  -  y^^).  n.    (a'"  +  c)(a"*  —  c). 

6.  (a2"  +  c2")(a'""  +  C"").  12.    (a  —  3  a63)(a  +  3  a^^). 

13.  (a -{- byx"" '  (a  —  2  b)xyp. 

14.  (m  —  n  —  2p)a;''  •  —  3  m^npx\ 

15.  (m^  —  n^)x*y^  •  (m^  +  7i^)x^y^. 

16.  (i?^  -  p  +  l)a^Y^  •  (i?  4-  l)a^2/"- 


REVIEW 

ORAL  EXERCISES 

State  the 

products : 

1.  4  a; 

3.  7  a 

5.    —9m 

7.   -2aa; 

9. 

7a;22/ 

9a; 

4a 

3m 

4-5a;2 

-     xy 

2.        82/ 

4.-6^ 

6.    5a6 

8.   —602/ 

10. 

—  9  am 

-32/ 

-2t 

7ac 

—  4:  ay 

^ar 

WRITTEN   EXERCISES 
Multiply  and  test : 


1. 

ax-^S 

ax-{-  5 

3.   2/+    «  +  & 
2/  +  5a;-& 

5.    1  — aj  +  a?* 
l+a;-ar^ 

2. 

4a6  +  c 
2  a6  +  3  c 

x^^Tx'  +  Bx- 
2x-4. 

4.    a-5- 
a  —  b- 

-c 
-c 

6.   «  +  3a;  — 1 

x^2a-\-l 

7. 

3 

9. 

x"^ -\- S  px  —  4:  p"^ 
2x'-l  px-p^ 

8. 

a3  _  3  aV  4-  3  a2/2 
a-2/ 

-2/' 

10. 

a^  +  Zx^y  +  ^xy'^  +  f 
x  +  y 

MULTIPLICATION  83 


11. 

x-5 

a;  +  6 

2a;  +  3 
x-1 

3aj  +  5 
2x-4. 

14. 
15. 
16. 

6t-3u 

t-\-2u 

3a;2_^5 
x-^ 

x^  +  x  +  1 
X  H-1 

17. 
18. 
19. 

x-1 

12. 
13. 

x'  +  3x  +  2 

a;2  _j_     a;  +  2 

2  a;2  —  a?  +  1 
2  a;  -5 

20.  (2a"  +  3  6«)(2a'*  +  3  6").  29.  o^yix^'y^-xof). 

21.  (?-p~^  —  sP-^)(r  —  s).  30.  (a  —  5)  a; .  (a  +  6)  a;. 

22.  (42;3"-22"+2"-l)(30"+l).  31.  a6.(c-l)6. 

23.  (3  x"^  —  y%3  a;"*  +  y').  32.  aa;^ .  (a  —  5)  x^  •  (6  +  c)  ar^. 

24.  (a;-l)(a;-2)(a;-3).  33.  (x'y -xy''){x-\-y), 

25.  (a;  +  6)(a;-5)(a;-3).  34.  {m'' ■\-w? +  l){m^ -^1). 

26.  (a3-a2-l)(a4.1).  35.  (^/^  +  2/ +  2)(2/3  -  1). 

27.  (m2-m  +  l)(m  +  l).  36.  (2/^  -  3  ?/ +  5)(/ 4- 10). 

28.  (a;4-a^  +  l)(a;2  4-6).  37.  (rs  -  rV)(rs  +  ?V). 

SUMMARY 

The  following  questions  summarize  the  definitions  and  pro- 
cesses in  this  chapter : 

1.  State  how  to  find  the  product  of  two  monomials. 

Sec.  110. 

2.  State  the  Law  of  Exponents  in  Multiplication. 

Sec.  112. 

3.  State  how  to  find  the  product  of  several  monomials. 

Sec.  113. 

4.  State  how  to  multiply  a  polynomial  by  a  monomial 

Sec.  116 

5.  State  the  Distributive  Law  of  Multiplication.     Illustrate 
this  law  in  symbols.  Sec.  117. 

6.  State  how  to  test  the  work  in  multiplication. 

Sees.  118,  119. 

7.  State  how  to  multiply  one  polynomial  by  another. 

Sec.  123. 


CHAPTER   IX 
DIVISION 

DIVISION    OF   MONOMIALS 

125.  Preparatoky. 

1.  41b.-^2  =  (  )lb.     4yd.-f-2  =  (  )yd.     4.y---2=()y, 

2.  6oz.^3  =  ()oz.     6ft.--3  =  (  )  ft.       6/-^2  =  ()/. 

3.  3ci  .(  )  =  6a26,  then  6a26-^3a  =  ? 

4.  8  &2 .  (  )  =  16  ab^  then  16  ab^-^8b''  =  ? 

5.  a'b  '{)  =  a?bcd,  then  o?bcd  ^o?b  =  ? 

126.  Division.  Division  is  the  process  of  finding  one  of 
two  factors  when  the  product  and  the  other  factor  are  given. 
Division  is  thus  the  inverse  of  multiplication. 

The  problem  18  a%'^c  ~  3  a^&^c  means  to  find  the  number  by  which 
3  a^^c  must  be  multiplied  to  produce  18  a%'^c. 
To  do  this  : 

3  must  be  multiplied  by  6  to  produce  18. 
a^  must  be  multiplied  by  a  to  produce  a^. 
6*  must  be  multiplied  by  h^  to  produce  W. 
c  multiplied  by  1  produces  c. 
Hence,  the  quotient  of  18  a^h'^c  -^  3  a^h^c  is  6  •  a  •  &3  •  1,  or  6  ah^. 

Test.  The  product  of  the  divisor  and  the  quotient  must 
equal  the  dividend. 

127.  Zero  cannot  be  used  as  a  divisor. 

This  may  be  seen  by  reference  to  Sec.  114. 
For,  0  .  3  =  0    and    0  •  5  =  0. 
Therefore,  0  •  3  =  0  •  5. 

Now  if  we  could  divide  both  numbers  by  0,  the  result  would  be  3  =  5, 
which  is  not  true. 

84 


DIVISION  85 

WRITTEN   EXERCISES 
Divide  and  test : 
1.   65  a^b^  IS  ab.  6.   120  abc-h  20  a. 


2.   ASx^y^l2xy. 

7.   ^  mv"^  ■ 

-j-mu 

3.    63  mV -T- 21  ?ftn. 

8.   igf^ 

■it. 

4.   96  a'b'-i- 12  ab. 

9.   f  Trr^H 

■r-  7rr\ 

5.    45pV-15pV- 

10.    ^Trd'- 

^i,^d 

11.   Find  the  numbers  to  fill  the  blanks : 

(1)               (2) 

(3) 

(4) 

(5) 

Dividend:     12  a;^           27  ab 

SS  ay 

42  p 

36  mw 

Divisor :          3  x            

7p 

9m 

Quotient :      3  b 

11a 

128.    The  equation : 

Dividend  =  Divisoi 

'  X  Quotier 

it 

remains  true  if  the  two  numbers,  dividend  and  divisor,  are 
multiplied  (or  divided)  by  the  same  number. 

In  other  vs^ords : 

The  quotient  is  not  altered,  if  both  dividend  and  divisor  are 
multiplied  or  divided  by  the  same  number. 

Canceling  may  be  used  as  in  arithmetic : 

2  6^ 

Thus,  8a5«^g#«=2  6^ 

'  4.ab^     ^  jtf 

Exponents  must  not  be  canceled.     E.g.  if  we  should  cancel  the  exponent 
2  from  the  exponent  6,  the  result  would  be  b^  instead  of  6*. 

Taking  out  factors  results  in  subtracting  not  dividing  exponents. 

129.   Law  of  Exponents.     Since   a^  =  a^ '  a^,  it  follows   by 

a^ 
dividing  both  members  by  a^  that  —  =  a'. 

Likewise,  from  a"*"*"'"  =  a"*  •  a*",  it  follows  by  dividing  both 
members  by  a*"  that =  a"*. 


86  A   HIGH   SCHOOL   ALGEBRA 

Accordingly,  the  exponent  of  any  letter  in  the  quotient  is  the 
difference  between  the  exponent  of  that  letter  in  the  dividend  and 
its  exponent  in  the  divisor. 

For  example : 

^  =  a5,  also  ^  =  a2«, 


a'  a" 

ORAL  EXERCISES 


Divide : 


1.  a'^aK  ^  Ua^b  ,„  b'''+^  ,^    65a^y^ 

2.  a''-^a\                    2  a                        b^  13  xy^ 

3.  610.^53^  ^  12  g^'  ^g  25  m^n  18  a6a^.?/^ 
4  IP  ^jji.  *  lOa^*'  *      5  m*    *  *     9abxyp 

5.  a^b^a\  10.  ^.  14.  ^\  18.   ^«'^^"' 

6.  a'b^-^a'^b. 


a2«  a;?/"  3  tt62 


7.    ni'^w?.         11.   7^— r--  15.    =^1^-.  19.   — — ^. 

130.   To  find  the  quotient  of  two  monomials : 

1.  Find  the  sign  of  the  quotient,  by  the  rule  of  signs.  (Sec.  77.) 

2.  Find    the    numerical    coefficient,   by   dividing   the   given 
numerical  coefficients. 

3.  Find  the  literal  part,  by  dividing  the  given  literal  parts. 

Thus,  in  18  a^ft -= — 3a&,  the  sign  is  — ,  the  numerical  coefficient  is 
18  -7-  3  or  6,  the  literal  part  is  a%  -^  db  or  a^  -,  .-.  the  quotient  is  —  6  d^. 

WRITTEN   EXERCISES 
Divide ; 

^     2a^cs?y  ^     lOo^^  a'^'b*'' 

—  2axy  '     —5ax^  '    a^'^b" 

»     —  abcx^y  _      10  mhi^  „     18  <^^^' 
—  cx                           —  ^  myv  3  o^b  ■ 

g     -  5  ab&  g     1.4  a^&  ^     24  a6c 

'    -  56c  *  '     .la'^b  '  '     -3b' 


i 


DIVISION 

10. 

1.5a^yz 
-bx'^y 

14. 

10  2n^n+4 

18. 

-16aV?/3 
4  2/ 

11. 

w?'n?p'^ 

15. 

-2a3a^ 

\ax^ 

19. 

270^2/^^^ 

-n-  -p 

-3^' 

12. 

—  a}  •  —be 

16. 

-Ip^q^i^t 

1 

20. 

16  &6ca;6 
-  4  Wcx^ 

13. 

-  35  b'c' 

17. 

a'^b^c'^x 
—  a}bcx 

21. 

54  m^p^x^ 
-  9  mp  V 

22. 

^,n+7^3n+6^2n+9 

26. 

- 

-18aj«+' 

ya+7 

-  a'^b^c' 

3 

r^ya 

^-3 

23. 

3  a'b'&  -^  -  abc. 

27. 

1 

^r^ 

^i 

7rr. 

24. 

2  .  5  a  V2/  -^  - 

5ax^y. 

28. 

i 

77^3 

-i 

TTd^ 

25. 

55aV--ll 

o?x. 

29. 

- 

-27 

a;^?/^; 

^2  --  3  a;2/22. 

87 


DIVISION   OF    POLYNOMIALS 

131.  Pkeparatory. 

1.  2)  4  bu.    2   pk.  2)  4  lb.    2   oz.  2)  4?4-2g 

()bu.  (  )pk.  ()lb.  ()oz.  ()?+(  )0 

2.  3)  9  ft.   6    in.  3)  9  mi.  6   rd.  3)  9  m  4- 6  r 

()ft.()in.  ()mi.()rd.  (  )m  +  (  )r 

Multiplication  Division 

Since     (1)  then,  (2) 

a  —  bd-\-^G 

m  m)ma  —  mbd  +  5  mc 

ma  —  mbd -\- 5  mc  a—bd-^5c 

132.  Accordingly,  to  divide  a  polynomial  by  a  monomial,  divide 
each  term  of  the  polynomial  by  the  monomial,  and  use  the  signs 
obtained  as  the  signs  of  the  quotient. 

WRITTEN    EXERCISES 

Divide  and  test : 

1.  (6a2  +  3a)--3.  4.    (12  a6  +  5  6) -- 6. 

2.  (12a6  +  46)~-4.  5.    (6  a2  +  3  a)-f-3  a. 

3.  (6a2-f-3a)H-a.  6.    (12  a?) +  4  6) --4  6. 


88  A  HIGH   SCHOOL   ALGEBRA 

7.  5  a^  —  4  a6  +  4  a  by  a.  9.    {x^y  4-  xy^)  -r-  xy. 

8.  a«-5a5  +  3a^  by  a2.  10.    {^x^y -Q  xy)-^S  xy. 

11.  25a2  +  10a6  4-5  62by  5. 

12.  27  o?h  -  9  aZ/2  +  9  a^ft^  by  9  ah. 

13.  (12  mil  +  27  m/i^p)  -f-  3  m7i. 

14.  (6  ar^2/  —  "^  ^^^  +  ^  ^y^;)  -j-  2  «. 

15.  4  <?/^  —  8  a^2/3  +  6  xy^  by  —  2  xy. 

16.  -3a2-H|a6-3ac  by  -fa. 

17.  x^y  —  3  ic^^/  -|-  9  a;i/2  by  3  xy. 

18.  (a;^?/  -  3  xY  +  9  a;/)  --  3  xy. 

19.  (aV-2a6c2  4-3ac3)^ac2. 

20.  a^c^  -  2  a5c2  +  3  a&  by  -  ac\ 

21.  (5a363-35a26V  +  2a6V)--5a6. 

22.  (2  m^v?  —  3  mn^  +  4  m^ii  —  li^)  ^3/1. 

23.  (aVi/  —  3  G^hx^y  +  3  ah^xy"^  —  o?Wxy^)-T-axy. 

24.  iC^'*  4-  4  a;9":y2  _|_  3  ^8ny2n  ^y  ^n^ 

25.  d"^^  +  «'"+*  +  a"''+^'  by  a^ 

26.  10  ^'^+12  ^  10  8n+9  ^  -LQ  6n+6  ^y  IQ  4n+3^ 

133.    To  Divide  by  a  Polynomial.     This  process  is  seen  best 
from  examples. 

1.   Divide  x"^  +  ^  xy  ■\- 2  y"^  hj  y  ■{•  x. 

Arrange  the  terms  of  the  divisor  Quotient 

and  the  dividend  according  to  the  x  +  2y 

powers  of   the   same  letter  {x  in      Divisoe  x  -\-  y)x^  +  S  xy  -\-  2  y^ 
this  example).  x'^  +     xy 

Divide  the  first^erm  of  the  divi-  2xy  +  2y^ 

den^y  the  first  term  of  the  divisor.  2xy  -\-2y^ 

The  resulT^m  this  exaitrpl^a;) 
is  the  first  term  of  the  quotient. 

Multiply  the  entire  divisor  by  this  term  and  subtract. 

Divide  the  first  term  of  the  remainder  by  the  first  term  of  the  divisor. 
The  result  (in  this  example,  2  y)  is  the  second  term  of  the  quotient. 

Multiply  the  entire  divisor  by  this  term  and  subtract. 

Continue  the  process  until  a  remainder  zero  is  reached. 

Cases  in  which  such  a  remainder  cannot  be  reached  are  treated  later. 


Solution 

2a2 

-    Sab  -    4  62 

&)6a3 
6a3, 

-  17  a^b  +  16  63 

-  8a26 

-  9a26 

-  9  a^b-^  12  ab^ 

DIVISION  89 

Test.  Multiply  the  quotient  by  the  divisor.  If  the  work 
has  been  correctly  done  (including  the  multiplication),  the 
result  will  equal  the  dividend.  Or,  substitute  arbitrary  values 
and  proceed  as  in  previous  cases. 

2.   Divide6a3-17a26  +  1663by3a-4&. 

Test  (Substitute  a=b  =  l) 
Divisor  x    Quotient  =  Dividend 

3  a  -  4  6)6  a3  -  17  a^b  +  16  63  (3  -  4)  (2  -  3  -  4)  =  6-17  +  16 

-l.(-5)=6 
6  =  5 

-  12  a62  +  16  68 

-  12  a62  +  16  68 


WRITTEN   EXERCISES 
Divide  and  test : 

1.  a'^-\-2ab-\-b^hj  a  +  b.  4.   x"^ -{- 4:  xy -\- 4:  y^  hy  x -^  2  y, 

2.  a2  +  3a  +  2  by  a  +  1.  5.   8  d^ +7  cd-\- 2  d^  by  d-\- 3  a 

3.  a;2-lla;4-30by  ic-5.         6.   6  a^-T  a-3  by  2  a-3. 

7.  2  oc^  —  xy  —  3  y^  by  X  -\-  y. 

8.  3a2  +  a6-262by  3a-26. 

9.  6m^-}-  mn  —  2  n^  by  3  m  +  2  n. 

10.  12y^-^19y-21by3y-\-7. 

11.  a^ -\-3  a'b +  3  ab^-^¥  by  a-\-b. 

12.  96  a"  -  4.  ab -15  b^  by  12  a- 5  b. 
^^3.  a^ +  3  a'b -\-3  ab^r{-b^  by  0^  +  2  ab-\-b\ 

14.  a^  +  a262  +  ¥  by  a^-ab-{-  b^ 

15.  0^3  -  2  x2  _  2  a;  +  1  by  a;  + 1. 

16.  3p2_ll5_8pbyp  +  5. 

17.  3a^-2a-1865by  a-5. 

18.  14.y' -13 y - 43 y^ -\-32  f  -^3  by  7 y^ - 5 y -3. 

19.  21a3-4a2-16-46aby  4a4-2-3a2. 

20.  -i±^.  21.   -'-'' 


1  -  a2  +  a^  a^  +  62 


90  A  HIGH   SCHOOL   ALGEBRA 

22. 
23. 
24. 
25. 


x^p  +  8  a^p  -f  16 

26           ^^V  +  1        . 

a;^  +  4 

""■   2xY-2xy  +  l 

9  m2  -f  12  mn  +  4  ^2 

07        «^  +  &^ 

3m  +  27i 

a'^-ah-{-  b^ 

a^  +  4  64 

28.         ^^-^^      . 
a2  +  a6  +  52 

a2  +  2  a6  +  2  6^ 

6a262_a63-12  54 

„       ^6_4^Y_^4^2^4__^6 

3a6  +  462 

^2_2,. 

30    «^-«^-ll' 

a3  + 16  o2  _  2  a  -  3 

-4a 

+  3 

6a^^  +  17x-3- 
•     2  x3  +  3  x2  - 

-19aj 
-4.x- 

-4^ 
-1 

32.   ^"  +  3a.Y- 

-  a^^?/' 

^-3/«_ 

134.  Remainders.  In  algebra,  as  in  arithmetic,  the  divi- 
sion may  not  be  exact.  That  is,  no  remainder  may  be  zero, 
however  far  the  division  is  carried. 

Thus,  in  the  example  at  the  right,  there  is 
a  remainder,  2.     The  division  might  be  con- 
tinued, the  next  term  of  the  quotient  being 
2 
- ;  but  it  is  customary  to  stop  as  soon  as  a  re- 

X 

mainder  is  reached  that  is  of  lower  degree 
than  the  divisor.  The  integral  part  of  the 
quotient  has  now  been  found  ;  it  is  called 

the  integral  quotient. 

ii 

By  using  the  fractional  form  to  indicate  the  division  of  the  remainder, 
the  result  of  the  above  division  may  be  expressed  thus : 

?i±i  =  a:3  -  a;2  +  X  -  1  +  -^ — 
x-^\  x  + 1 

The  right  member  of  the  equation  is  called  the  complete  quotient. 

Test.   Dividend  =  divisor  x  integral  quotient-!- the  remainder. 

Substituting  %  —  \. 

Dividend  =  Divisor  x  Integral  Quotient  -h  Remainder. 
1  +  1  =  (1  +  1)(1-1  f  1-1) +2. 
2=2.0-H2. 


a:3  _  x2 

4- a.-. 

-  1 

a:+l)x4-Hl 

x!^^x^ 

-x^ 

+  1 

-x^ 

-x^ 

3;2 

+  1 

a-2 

-\-x 

-x+l 

-x-1 

DIVISION  91 

WRITTEN    EXERCISES 

Find  the  quotient  and  remainder : 

1.  (a^-l)-^(x  +  l).                  5.  (Sx^-5x  +  2)-i-(x-4:). 

2.  (a'-^x')^(a  +  x).                 6.  (Sf +  7  y -l)^(2y+3). 
x^-{-x  +  l                               „  6ax  —  9ay  —  4:bx-^Sby^ 

'      x^-1     '                                '  3  a -2  b 

x^-2x''  +  l  2a3-2a'^-6a4-4 

a;2-fl       *                            •  2a-3 


REVIEW 

ORAL  EXERCISES 

Divide  and  test : 

-  32  ax*  l_i6y  +  642/'             mY-n'^ 

'       4.x'     '  1-Sy        ' 

2     -42my^  ^    a^±b^^               ^^ 

6  m^?/  x^ 

2     144  gV  g    a^  —  y^^                   ^^ 

— 16  acc^  a?  —  2/ 

4     -14641  pV  g_  a:^-/_                  j4 

—  lip     '  '    x  +  y' 

5.  tjzMl+J^.        10.  5^^.  15. 

^  —  7  a  +  x  x'''  —  y 

16.   (14  aa^  +  6  a£c) -^ 2  a.  17.   (25^^-40  0^-5. 

18.  (lSx'z'-24:a^z')^-6xz'.. 

19.  (a^^c  -  ab^c  +  2  a&c^)  -^  a6c. 

20.  (—27  m^^i/^"  —  9  m^V)  -^  ^  m*2/". 

21.  (-  25i92a^3r  _|_  10J)3af^2r^)  h-  5  j^^^'". 

WRITTEN   EXERCISES 
Find  the  quotient  and  remainder,  if  any : 

1.  2)"  —  16  by  p  +  2. 

2.  a'  -  10  a2  +  25  by  a^  -  5. 

3.  6  2/2  +  2  y  -  20  by  2  2/  +  4. 


mp  +  ng 

X*- 

■y' 

x'- 

■f 

m*- 

-n' 

m^  ■ 

■fn^' 

a^y' 

—  m^n^ 

xy 

—  mn 

^,n. 

-/' 

92  A   HIGH   SCHOOL  ALGEBRA 

4.  2  a2  +  5a6H- 2  62  by  a +  2  6. 

6.  6a^b^ -ab''-12b' by  2ab-3b\ 

6.  x^  -\-  ax-^bx  +  ab'hj  x -{- a. 

7.  m'^  —  m-p'  +  m-p^  —p^  by  7n^  —p^. 

8.  i«3-f-6a:2  +  8a;-3by  x2-j-3x-l. 

9.  2ac  — be  — 6a^-\-3ab  by  c  — 3  a. 
10.  1  -  a;2  +  2  a:^  —  x"^  by  a;^  —  a;  +  1. 

a^-1 


11. 
12. 
13. 
14. 
19. 


x^-l 

x-3 

g^  -  5  g^  4-  4 

a4-2 
a^^2x^  +  x-4: 

x-3  '  ~"  7'^-5r  +  2 

18  a;^  -  24  a;^  4-  38  ar^  -  68  a;  +  32 
3a;-2 


15. 

x'-2x'-^3x 
^-1        ' 

1  ft 

8_12a  +  6o2-a3 

2-a 

17. 

15a'b  +  6ab'-\-Sa'-h3b^ 

Sa-b 

1Q 

-7r'  +  37^-^5  +  r* 

Divide : 

20.  ^-1  by  z-1. 

21.  x^  —  oc^y^  hy  X  —  y. 

22.  a*-2a2  +  l  by  a  —  1. 

23.  a^  +  2ab-\-b^  —  c^  by  a-\-b-{-c. 

24.  m  +  -   by   — (-i>  to  four  terms. 

p  m  , 

25.  X*  +  x^y^  +  y*  by  x^  -\-y^  —  xy. 

26.  i  +  l  +  g^  by  V-  +  1- 

27.  a:2  — 2/2  — 2?/  — 1  by  a;  +  2/  +  l. 

28.  6a;2  — 2a;2/  — 3a;  +  2/  by  3  a;  — ?/. 

29.  a;3m  _|_^3«  13^  a^^i-f.^/™. 

30.  6  r«  -  18  r«  4- 18  r*  -  3  r^  -  9  r^  +  9  r  -  3  by  2  7^  4- 1. 

31.  a^^  4-  4  a3^5  +  6  a^^^s  ^  4  ^^53  4.  54  ^y  a2x  ^  2  a'^ft  4-  h\ 


DIVISION  93' 

SUMMARY. 

These  questions  summarize  the  definitions  and  processes  in 
this  chapter: 

1.  State  in  the  form  of  an  equation  the  relation  between  the 
dividend,  divisor,  and  quotient.  Sec.  128. 

2.  State  the  Law  of  Exponents  in  division.  Sec.  129. 

3.  State  how  to  find  the  quotient  of  two  monomials. 

Sec.  130. 

4.  Explain  how  to  divide  a  polynomial  by  a  monomial. 

Sec.  132. 

5.  State  how  to  divide  by  a  polynomial.  Sec.  133. 

6.  State  how  to  U.'-t  the  work  of  division.        Sees.  126,  133. 

7.  When  is  the  remainder  reached  in  the  process  of  division  ? 

Sec.  134. 

HISTORICAL  NOTE 

After  solving  many  exercises  in  the  four  processes  with  polynomials, 
algebra  may  seem  to  consist  mostly  of  such  manipulations  ;  but  we  shall 
see  as  we  proceed  that  the  solution  of  equations  becomes  more  and  more 
prominent  and  that  the  facility  we  have  gained  in  the  use  of  symbols 
is  a  help  in  handling  equations.  In  fact,  the  word  "algebra"  itself  is 
connected  in  its  origin  with  the  equation.  There  lived  in  Bagdad,  Arabia, 
about  825  a.b.,  a  famous  mathematician  known  froin  his  birthplace, 
Kharezm,  as  Al-khowarazmi,  and  in  the  title  of  his  work  signifying  "The 
science  of  transposing  and  combining  in  solving  equations,"  there  appeared 
the  Arabic  word  Al-Jahr  to  denote  these  processes.  When  this  manu- 
script was  translated  into  Latin  in  the  thirteenth  century,  the  word 
Al-Jahr  was  transferred  as  algehrce,  from  which  we  have  our  word 
"algebra."  Thus,  algebra,  instead  of  meaning  originally  "science 
of  symbols,"  as  we  might  imagine,  means  "  the  science  of  processes 
in  equations," 

Although  Diophantos  (300  a.b.)  could  find  the  product  of  two  bino- 
mials, the  knowledge  of  how  to  multiply  or  divide  one  polynomial  by 
another  is  comparatively  recent.  Ordinary  algebraic  division  was  devel- 
oped in  the  seventeenth  century,  and  Newton,  in  his  Arithmetica  Uni- 
versalis (1707),  showed  how  arranging  the  terms  in  both  dividend  and 
divisor  according  to  the  descending  powers  of  the  same  letter  facilitated 
the  work. 


CHAPTER   X 

EQUATIONS 

135.   The  parenthesis  is  often  used  in  equations  to  indicate 
multiplication. 

EXAMPLES 


1.    Solve: 

(x  +  l)(x-5)=^x(x-l). 

(i) 

Removing  the  parentheses,          x"^  —  4:X  —  6  ==  X^  —  X. 
Then,                                                           -  3  X  =  5, 
and                                                                    X  =  —  f . 
Test.                      («  5  +  i)(^_  5  _  5)  ,,,  4^  ,,_  5  (_  |  ^  1). 

2.    Solve: 

(ax  —  l)x  +  2  —  b  =  ax^  -  c. 

(^) 

Removing  the 
Uniting  terms 

parenthesis,     ax^  —  X  +  2  —  &  —  ax'^  —  C. 
-x  +  2-b=-c. 

(2) 

(3) 

7\2 


Then,  -x=-c  +  b  -2, 

and  -  x  =  C-h  +2.  (4) 

Test  by  substitution. 


WRITTEN   EXERCISES 

Solve : 

1.  x(:x-S)  +  l-x(x-5)=0.  8.  {x-iy=(x-sy. 

2.  x''-[-3-x(x  +  4)=:  15.  9.  (x  -5)(x-{-S)  =  (x-  7) 

3.  x\x-l)-^+x''-2x=12.  10.  y(9y-5)  =  (Sy-\-iy. 

4.  5(0^-4)  =  6 (aj  +  1).  11.  e-l  =  (t-^4:y. 

5.  _3(a;  +  7)  =  2(l-3x).  12.  15|)  =  - 29  -  (4  -  4i>). 

6.  a(x-b)  =  Sab.  13.  1  -  (16 +  7  w;)  =  8  m;. 

7.  (x-4:)(x-\-4:)  =  x--Sx.  14.  2  a; -(7  a; -18)  =  4  a;. 

15.  (a;  +  5)(2a;-l)  =  a;(2a;  +  4). 

16.  (2a;  +  i;(2a;-l)  =  a;(4a;-2). 

17.  3x{6xi-5)  =  18  x^  -  (a;  +  32). 


94 


EQUATIONS  95 

18.  4y(y-l)  =  (2  2/-l)(2  2/  +  l). 

19.  4a;  +  8  =  2-(9x  +  20).         23.    ax(x - 5)  =  ax" - 1, 

20.  4s  =  7— (s  — 3).  24.  (a  +  b)x  =  cx-{-5. 

21.  (x-\-2){x  —  5)  =  x{x  —  l).     25.  (x  —  l)(ax  +  b)  =  ax^  +  b. 

22.  (ca;  +  2) a?  =  ca?^  —  a.  26.  (m +  l)a;— />«  =  g'. 

27.  p(a?  — 5)4-4p  =  l. 

28.  (x  -  1)  (x  +  2)  =  (a;  -|>)  (a;  +  g). 

29.  (aa;+6)(c  +  c^)=/. 

30.  (aa;  —  1)  (bx  —  1)  =  (ax  +  b)  (bx  —  c). 

136.  Problems  often  result  in  equations  in  which  paren- 
theses may  be  used. 

EXAMPLES 

1.  Express  by  an  equation:  2%  of  (100  —  x)  dollars  equals 
$1.40. 

2.  Express  by  an  equation :  4  of  the  quantity  75  —  2  a; 
equals  44. 

3.  The  difference  between  twice  a  number  and  the  number 
less  10  is  22.     Express  this  fact  by  an  equation. 

4.  A  man  has  $  500  of  which  x  dollars  is  in  the  bank  draw- 
ing interest  at  4%,  and  the  remainder  is  lent  at  6%.  Express 
the  annual  interest  received  on  the  $500. 

5.  The  amount  of  $100  for  one  year  at  5%  simple  interest 
is  (1  +  .05)  100  dollars.  Express  the  amount  of  x  dollars  at  ?*% 
for  1  yr. ;  for  6  yr. 

WRITTEN    EXERCISES 

Solve  and  test : 

1.  A  man  had  $500,  of  which  he  invested  x  dollars  at  4% 
per  annum,  and  lent  the  remainder  at  6%  per  annum.  His 
annual  interest  was  $28.  Find  how  many  dollars  were  in- 
vested at  4%  and  how  many  were  lent  at  6%. 

2.  Find  the  principal  that  will  amount  to  $32.70  at  the 
rate  of  4^%  per  annum  for  two  years. 


96  A  HIGH  SnHOOL   ALGEBRA 

3.  The  amount  of  a  certain  principal  at  4  %  simple  interest 
for  2i  years  is  $  220.     What  is  the  principal  ? 

4.  The  amount  of  a  certain  principal  at  5^  %  simple  inter- 
est for  21-  years  is  $  284.375.     What  is  the  principal  ? 

5.  $  100  is  divided  into  two  parts,  of  which  one  is  x  dollars. 
What  is  the  other  ?  If  the  first  part  exceeds  the  second  by 
$  10,  find  each  part. 

6.  $600  is  divided  into  two  parts,  one  of  which  is  2  a?. 
Write  an  expression  for  the  other.  The  first  part  equals  ^ 
the  second.     Find  each  part. 

7.  The  total  amount  of  insurance  in  force  in  New  York 
City  and  Buffalo  in  a  recent  year  was  $2,700,000,000;  Buffalo 
had  -f^  as  much  as  New  York.     How  much  had  each  ? 

Solve  Exercises  8-12  by  equations  requiring  the  use  of  the 
parenthesis :  Then  solve  each  by  an  equation  not  requiring  a 
parenthesis : 

8.  Japan  recently  gave  American  manufacturers  an  order 
for  2000  cars  and  locomotives  ;  it  consisted  of  19  times  as  many 
cars  as  locomotives.     How  many  of  each  were  ordered  ? 

9.  The  amount  of  condensed  milk  produced  by  New  York 
and  Illinois  is  f  of  that  produced  by  the  rest  of  the  country ; 
the  total  amount  produced  in  the  country  annually  is  about 
154  million  pounds.  How  many  pounds  are  produced  by  the 
two  states  together  ? 

10.  The  United  States  produces  3  times  as  much  cotton  as 
the  rest  of  the  world ;  the  total  cotton  production  in  a  recent 
year  was  14  million  bales.  What  was  the  number  of  bales 
produced  by  the  United  States  ? 

11.  Mississippi  and  Texas  together  produced  4  million 
bales  ;  Texas  produced  If  times  as  much  as  Mississippi.  How 
many  bales  did  each  produce  ? 

12.  The  United  States  consumes  f  as  much  cotton  as  does 
the  rest  of  the  world.  How  many  bales  is  this,  when  the 
whole  world  consumes  14  million  bales  annually  ? 


EQUATIONS  97 

137.  The  use  of  the  parenthesis  often  makes  it  possible  to 
reduce  the  number  of  unknowns. 

EXAMPLE 

The  sum  of  two  numbers  is  20,  and  3  times  one  of  them  less 
4  times  the  other  is  4.     Find  the  numbers. 

This  problem  suggests  at  once  two  unknowns,  but  it  can  readily  be 
solved  by  the  use  of  one  unknown,  and  practice  in  this  work  is  good 
training  in  mathematics. 

1.  Let  X  be  one  of  the  required  numbers. 

2.  Then,  20  -  x  is  the  other. 

3.  3  a;  —  4  (20  —  x)  =  4,  by  the  conditions  of  the  problem. 

4.  Therefore,  7  x  -  80  =  4,  and  x  =  12. 

6.   Then,  20  —  x  =  20  —  12,  or  8,  and  the  numbers  are  12  and  8. 

The  use  of  the  parenthesis  in  step  (3)  takes  the  place  of  a  second  un- 
known. 


\ 


WRITTEN   EXERCISES 

Solve  and  test : 

1.  The  sum  of  two  numbers  is  25,  and  twice  one  of  them 
plus  3  times  the  other  is  60.     What  are  the  numbers  ? 

2.  The  sum  of  two  numbers  is  30.  ^  of  one  of  them  less  J 
of  the  other  is  3.     What  are  the  numbers  ? 

3.  The  sum  of  two  numbers  is  38.  One  of  them  less  |  of 
the  other  is  13.     What  are  the  numbers  ? 

4.  The  sum  of  2  numbers  is  20.  2  times  the  larger  number 
less  3  times  the  smaller  is  5.     Find  the  numbers. 

5.  The  sum  of  two  numbers  is  42.  When  the  larger  num- 
ber is  diminished  by  5,  ^  the  result  is  the  smaller  number 
less  7.     Find  the  numbers. 

6.  The  product  of  two  consecutive  whole  numbers  dimin- 
ished by  the  square  of  the  smaller  is  29.     Find  the  numbers. 

7.  The  product  of  two  consecutive  whole  numbers  less  the 
square  of  the  smaller  is  49.     Find  the  numbers. 

8.  The  difference  between  two  numbers  is  2,  and  their 
product  diminished  by  the  square  of  the  larger  is  ~  16.  Find 
the  numbers. 


98  A  HIGH   SCHOOL   ALGEBRA 

138.  Certain  problems  of  measurement  are  best  solved  by- 
equations  requiring  the  use  of  the  parenthesis. 

EXAMPLES 

1.  The  inside  measurements  of  a  rectangular  garden  are 
20  ft.  by  30  ft.  The  outer  margin  is  li  ft.  wide.  What  are  the 
outside  measurements  of  the  garden  ?  Express  its  area.  If  the 
margin  were  x  ft.  wide,  express  the  area  of  the  garden. 

2.  Express  the  outside  measurements  of  a  rectangular  gar- 
den consisting  of  a  walk  2  a  ft.  wide  around  an  inner  plot  50  ft. 
by  70  ft.     Express  its  area. 

3.  Express  the  difference  between  the  area  of  a  square  x  ft. 
on  a  side  and  the  area  of  a  square  {x  +  1)  ft.  on  a  side. 

4.  Express  the  difference  between  the  area  of  a  square 
(a?  +  2)  ft.  on  a  side,  and  the  area  of  a  square  formed  by 
diminishing  this  length  by  5  ft. 

WRITTEN    EXERCISES 

Solve  and  test  •. 

1.  The  difference  between  the  area  of  a  square  x  ft.  on  a  side 
and  the  area  of  a  square  {x  -j-  1)  ft.  on  a  side  is  13  sq.  ft.  Find 
the  side  of  the  first  square. 

2.  The  difference  between  the  area  of  a  square  {x  -f  2)  ft.  on 
a  side,  and  the  area  of  a  square  formed  by  diminishing  this 
length  by  5  ft.  is  75  sq.  ft.     Find  the  side  of  each  square. 

3.  The  inside  measurements  of  a  picture  frame  are  10  in.  by 
14  in.  The  width  of  the  frame  is  a;  in.  What  are  its  outside 
measurements  ?  If  the  area  of  the  frame  less  4ar^  is  120  sq.  in., 
what  is  its  width  ? 

4.  The  area  of  a  square  of  side  x  equals  the  area  of  a  rec- 
tangle, one  of  whose  sides  is  a?  —  6,  and  the  other  x  -f  12. 
Find  X. 

5.  The  area  of  a  square  equals  the  area  of  a  rectangle,  one 
of  whose  sides  exceeds  the  side  of  the  square  by  10  in.,  and  the 
other  is  less  than  the  side  of  the  square  by  6  in.  Find  the 
dimensions  of  each  figure. 


EQUATIONS  99 

Problems  of  Motion : 

1.  If  an  electric  car  moves  at  the  rate  of  1^  blocks  per 
minute,  how  far  will  it  move  in  4  min.  ?  If  it  moves  at  the 
rate  of  x  miles  per  hour,  how  far  will  it  move  in  (m  +  1)  hr.  ? 

2.  Let  d  equal  the  distance  traveled  in  t  hr.  at  the  rate  of 
r  mi.  per  hour.     Express  d  in  terms  of  r  and  t. 

3.  Solve  the  equation  d  =  rt  for  t ;     solve  it  for  r. 

4.  In  d  =  rt,  find  d  when  t  =  a  +  1  and  r  =  6  —  1.  Find  r 
when  d  =  x'^  —  y^  and  t  =  x  —  y. 

6.  d  =  ?t  is  the  equation  for  distance  when  a  body  so  moves 
that  its  rate  may  be  taken  as  uniform.  If  sound  travels  1100 
ft.  per  second,  how  far  away  is  a  gun  when  the  report  of  firing 
is  heard  3J  sec.  after  it  occurred? 

6.  A  fort  is  10  mi.  away.  According  to  Exercise  5,  how 
long  after  firing  will  it  be  before  the  report  is  heard  ? 

7.  If  A  travels  x  mi.  an  hour  and  B  follows  A  at  the  rate 
of  y  mi.  an  hour,  going  faster  than  A,  express  the  number  of 
miles  that  B  gains  on  A  per  hour.  Express  how  long  it  will 
take  B  to  gain  d  mi. 

8.  If  A  travels  x  mi.  an  hour  and  has  c  hr.  the  start  of 
B,  how  far  ahead  is  he  when  B  starts  ?  According  to  Exer- 
cise 7,  in  how  many  hours  will  B  overtake  A  ? 

9.  An  automobile  leaves  city  A  at  7  a.m.,  going  20  mi.  an 
hour.  A  motor  cycle  follows  2  hr.  later,  going  25  mi.  an 
hour.  How  far  ahead  is  the  automobile  at  9  a.m.  ?  How  far 
from  A  will  the  motor  cycle  overtake  tJie  automobile? 

10.  A  steamer  leaves  its  dock  and  travels  16  mi.  per  hour. 
It  is  followed  30  min.  later  by  a  motor  boat  traveling  20  mi. 
per  hour.  How  far  will  they  be  from  the  dock  when  the  motor 
boat  overtakes  the  steamer  ? 

11.  Two  cyclists  A  and  B  start  at  the  same  time  from  M 
and  P  respectively,  100  mi.  apart,  and  travel  toward  each  other, 
A  at  the  rate  of  15  mi.  per  hour  and  B  at  the  rate  of  20  mi. 
per  hour.  How  many  hours  after  starting  do  they  meet,  and 
how  far  from  M  ? 


100  A  HIGH  SCHOOL   ALGEBRA 

REVIEW 

WRITTEN   EXERCISES 

1.  If  C  has  X  dollars  and  B  has  half  as  much,  express  what 
they  both  have.  If  this  amount  is  $75,  write  and  solve  an 
equation  that  will  find  what  each  has. 

2.  Express  the  area  of  a  rectangle  whose  length  is  n  ft,  and 
whose  width  is  4  ft.     If  the  area  is  48  sq.  ft.,  find  n. 

3.  The  width  of  a  rectangle  is  x  ft.  and  its  length  is  twice 
the  width.  Express  its  perimeter.  If  the  perimeter  is  24  ft., 
find  the  length  and  width  of  the  rectangle. 

4.  Express  the  bank  discount  on  m  dollars  for  4  mo.  at  6%. 
If  the  discount  is  $  3,  find  m. 

5.  Express  the  interest  on  s  dollars  at  6%  for  Syr.  If  this 
interest  is  $  36,  find  s. 

6.  Express  the  area  of  a  triangle  whose  base  is  8  in.  and 
whose  altitude  is  a  in.     If  its  area  is  24  sq.  in.,  find  a. 

7.  A  merchant  marked  an  article  d  dollars  and  sold  it  at  a 
10%  discount.  Express  the  selling  price.  If  the  article 
brought  $7.20,  find  d. 

8.  In  how  many  years  will  $300  yield  $108  at  6%  inter- 
est ? 

9.  The  perimeter  of  a  rectangle  is  30  ft.  If  the  length  of 
the  base  is  twice  the  altitude,  find  the  area  of  the  rectangle. 

10.  A  certain  number  plus  twice  the  same  number  is  51. 
Eind  the  number. 

11.  What  number  added  to  3  times  itself  equals  64  ? 

12.  Divide  the  number  21  into  three  such  parts  that  the 
first  is  twice  the  second  and  the  second  is  twice  the  third. 

13.  If  a  certain  number  is  multiplied  by  12,  the  product  is 
168.     Find  the  number. 

14.  A  man  sold  a  quantity  of  wood  for  $  49,  half  of  it  at  $3 
a  cord  and  the  other  half  at  $  4  a  cord.  How  many  cords  of 
wood  did  he  sell  ? 


EQUATIONS  lOi 

15.  The  sum  of  the  ages  of  a  father  and  son  is  42  yr.  and 
the  father  is  five  times  as  old  as  the  son.  What  is  the  age  of 
each  ? 

16.  Two  men  start  from  the  same  place  and  travel  in  oppo- 
site directions,  one  35  mi.  a  day  and  the  other  25  mi.  a  day. 
In  how  many  days  will  they  be  360  mi.  apart  ? 

17.  Two  men  start  from  the  same  place  and  travel  in  the 
same  direction,  one  35  mi.  a  day  and  the  other  25  mi.  a  day. 
In  how  many  days  will  they  be  360  mi.  apart  ? 

18.  A,  whose  horse  travels  at  the  rate  of  10  mi.  an  hour, 
starts  2  hr.  after  B  from  the  same  place.  If  B's  horse  travels 
at  the  rate  of  8  mi.  an  hour,  how  many  miles  must  A  drive 
in  the  same  direction  to  overtake  B  ? 

Suggestion.  Let  x  equal  the  number  of  hours  traveled  by  A  before 
he  overtakes  B. 

19.  A  flag  pole  105  ft.  high  was  broken  so  that  the  length 
of  the  part  broken  off  was  six  times  the  length  of  the  part  left 
standing.     Find  the  length  of  each  part. 


CHAPTER  XI 
TYPE   PRODUCTS 

139.  Certain  products  are  specially  important  because  they 
serve  as  types  or  models  for  other  multiplications.  They 
apply  to  positive  and  negative  numbers  alike. 

140.  Type  I :  jr(/  -f  z)  =  jr/  +  xz. 
Type  II :          x(y  —  z)  =  xy  —  xz. 

For  example : 

a(J)  4-  c)  =  a6  +  ac. 

5  a;(3  —  y)  =  \bx  —  b  xy. 

2  a\a  -  5  6)  =  2  a^  -  10  a%. 

-  3  a6(c2  -  4  a(Z  +  6)  =  -  3  ahc"^  +  12  a%d  -  3  a&2. 

WRITTEN   EXERCISES 

Multiply : 

1.  -  x{x  -\-y).  4.  cx(w  +  z).  7.   4  ah(a  +  2  h). 

2.  c(a  —  6).  '  5.  —  y{x  —  y).  8.   5  a.-?/(a)2  —  2/2). 

3.  a(i  +  ^2),  6.  t(u  -f  i  a^).  9.  pg(m  —  n). 

10.  —  2  a;(3  ic^  —  2  072/).  l*^-    (5  ic  —  acy)(—  acxy). 

11.  -  3 a^6«x'(a2  -  62).  is.    (3  A- 8ctic3)(_  3  ^3^^^ 

12.  2  xy{\  xY  —  1).  19.  (6  amP  +  2  6^i«)(  -  6  m^^z) 

13.  -  4  0^2(3 ^_  22/).  20.  (9  a'¥  -  3  cd^){- abed). 

14.  2  m2(m  —  ^2).  21.  ^(2/  +  2;  +  w;). 

15.  3  2/(4  a; -2/).  22.  -3  a&(a2_  52+ c^). 

16.  (5  a2  -  4  62) (_  a252),  23.    7r(r,2  +  ^f  +  rirg). 

102 


TYPE  PRODUCTS 


103 


141.  Type  III:    (jr +/)2  =  x^  +  2  jr/ +/. 
In  words : 

The  square  of  the  sum  of  two  numbers  is  the  square  of  the  first, 
plus  twice  the  product  of  the  first  and  second,  plus  the  square  of 
the  second. 

142.  Type  IV :    (jr  -yY  =  x^  -  2  xy  +/1 

In  words : 

The  square  of  the  difference  of  two  numbers  is  the  square  of  the 
first,  minus  twice  the  product  of  the  first  and  the  second,  plus  the 
square  of  the  second. 

For  example  : 

(a  +  &2)2  :^  «2  +  2  a&2  +  64. 
(2  a-  6)2  =  (2  a)2  -  2(2  a)b  +*  b^ 
=  4  a2  -  4  a&  +  &2. 
232  =  (20  +  3)2  =  202  +  2  .  20  .  3  +  32 
=  400  +  120  +  9  =  529. 

Evidently  the  above  types  include  expressions  either  of  the 
form  (ax  +  6)^  or  (ax  —  by. 


WRITTEN   EXERCISES 


Square  as  indicated : 


1. 

(n  +  uy. 

13. 

(x+iy. 

25. 

{mn  +  u'^y. 

2. 

(x'-\-y'y. 

14. 

(2x-^iy. 

26. 

(2  ab  H-  6c)2 

3. 

{a  +  Sby. 

15. 

{2x'-\-iy. 

27. 

{abc  + 1)2. 

4. 

(w  +  2  7iy. 

16. 

(2a;2+3  2/2)2. 

28. 

332. 

5. 

{3x-{-2yy. 

17. 

252  or  (20  +  5)2. 

29. 

522. 

6. 

(a +  2)2. 

18. 

412. 

30. 

912. 

7. 

(a'-\-iy. 

19. 

822. 

31. 

172.    • 

8. 

{a -by. 

20. 

762  or  (80  _ 

4)2. 

32. 

972. 

9. 

(x-iy. 

21. 

(a2-62)2. 

33. 

462. 

10. 

(2x-iy. 

22. 

(3  a- 2  6)2. 

34. 

892. 

11. 

i^-W- 

23. 

{t~uy. 

35. 

112. 

12. 

(t-^wy. 

8 

24. 

iahc  -  1)2. 

36. 

362. 

104  A  HIGH  SCHOOL  ALGEBRA 

143.    Trinomials  may  also  be  squared  by  Types  III  and  IV. 
For  example  : 

(a  +  5  +  c)2  =  (a +  6  +  c)2  =  (a  +  by  +  2(a  +  b)c  +c^ 

=  a2  +  2  a6  +  62  +  2  ca  +  2  &c  +  c-2 
=  a2  +  62  +  c2  4-  2  a&  +  2  6c  +  2  ca. 
(2x-y  +  zy2=(2x-  yy  +  2(2  x  -  y)z  +  z^ 

=  4:  x^  —  i  xy  +  y^  +  i  xz  —  2  yz  -\-  z'^. 

In  words : 

TJie  sqy^'e  of  a  pqlyngjmQl  is  the  sumyf  the  squares  of  each 
of  its  terms  andUvice  the  product  of  every  two. 

WRITTEN    EXERCISES 

Square  the  trinomials  as  indicated : 


1.  (a  -\-b  —  cy.  4.    (x  —  y-\-  zf.  7.    {mn  -\-pq+  rsy. 

2.  {2a-b^)\         5.    {x  +  w-2z^)\       8.    (1-6  2/  +  /)^ 

3.  (a-'dh-  cf.         6.    iia-lb-^lcy.    9.    {m'-hmp-q'y. 

144.     Type  V :  (jr  4-/)(jr  -/)  =  x' -  /. 

In  words : 

The  product  of  the  sum  and  the  di^erence  of  two  numbers j^s 
the  difference  of  their  squares. 

For  example  : 

(a4-6)(a-6)  =  a2  -  62. 
(2  a  +  6)  (2  a  -  6)  =  (2  ay  _  62  =  4  a2  _  52. 
(m2  +  7i2)(m2  -  w2)  =  (m2)2_  (%2)2  =  wi*  -  n*. 

Evidently  the  above  type  includes  expressions  of  the  form 
(ajr  +  6)(ajr-6). 

ORAL    EXERCISES 

Multiply : 

1.  {in  -  n){m -\- n).     4:.{t +  u)t{  —  u).  7.    {x -\-b){x  —  b). 

2.  {a-x){a^x).        5.    {x-l)ix  +  l).       8.    {2x-l)(2x+l). 

3.  (p-q){p  +  q).       6.    (a;  -  2)(ir  +  2).       9.    (2a; -2/)(2a;+^). 

10.  (l+^-')(l-^).  13.    (2  a^  4- 3)(2  a2  -  3). 

11.  (ax -{- by)(ax  -  by).  14.    (a^^ +  b){a^  -  b). 

12.  (2a;-3  2/)(2a;  +  3  2/).         15.    (a^  -  3  ax)(a^  +  3  ax). 


TYPE   PRODUCTS  105 

16.  (ax-a^)(ax-^x').  17.    (i  x- ^y)  (^x  i-y). 

18.  (a  —  x) (a  4-  x)(a'^  4  a;^). 

19.  (a-x)(a-^x){a'^-{-x'^)(a'^-\-x*). 

20.  (l-r)(l  +  r)(l+r2)(l  +  r^)(l  +  r8). 

21.  (l-r)(l  +  r)(lH-r2)(l  +  r4)(l+r8)(l  +  ri6). 

145.  Two  numbers,  one  greater  than  a  multiple  of  10,  and 
the  other  less  than  this  multiple  by  the  same  amount,  may  be 
multiplied  according  to  Type  V. 

Thus,  93  .  87  =  (90  +  3) (90  -  3)  =  90"^ -  3"^  =  8100  -  9=  8091. 

WRITTEN   EXERCISES 

1.  31.29  =  (30  +  1)(30-1)  =  ?  3.   35-45  =  ? 

2.  42.38  =  (40  +  2)(40-2)  =  ?  4.   57-63  =  ? 

5.  21  .  19  =  ?  9.   44  .  36  =  ?        13.   99  •  101  =  ? 

6.  32  .  28  =  ?  10.   91  .  89  =  ?        14.   98  •  102  =  ? 

7.  29  .  31  =  ?  11.   53  .  47  =  ?        15.   90  •  110  =  ? 

8.  66.54  =  ?  12.    16.24  =  ?        16.    127-113  =  ? 

17.  What  is  the  cost  of  21  doz.  eggs  at  19  j^  a  dozen  ? 

18.  What  is  the  cost  of  28  lb.  of  butter  at  32  ^  a  pound  ? 

19.  How  many  oranges  in  146  crates  of  154  oranges  each  ? 

20.  What  is  the  area  of  a  rectangle  whose  dimensions  are 
62  ft.  and  58  ft.  ? 

21.  How  far  does  a  train  travel  in  37  hr.  at  the  rate  of  43 
mi.  per  hour  ? 

146.  Type  VI :  (jr  -|-  a)(jr  +  6)  =  jr^  +  (a  +  6)  jr  -f  ab. 
For  example : 

(x  +  5)(x4-3)=x2  +  8ic+ 15. 
(jc  -  3)  (x  +  7)  =  x^  +  4  a;  -  21. 

(3 X  +  c) (3 a; -  d)  =  (3 xY  +(0  -  d)Sx- cd  =  9x^  +  S(c-  d)x  -  cd. 
91-87  =(100  -  9)(100  -  13)=  100^  -  22  •  100  +  9  -  13 
=  10000  -  2200  +  117  =  7917. 

Evidently  this  type  includes  expressions  of  the  form 
(ajr  +  6)(ajr  +  c). 


106  A  HIGH  SCHOOL  ALGEBRA 

ORAL  EXERCISES 

State  the  products : 

1.  (0^  +  2)  (0^  +  5).  6.  (4^ -5)  (4^ -9). 

2.  (a  +  3)(a  +  6).  7.  {p  +  q){p  +  2q). 

3.  (6-5)(64-2).  ,  8.  (l-x){l-y). 

4.  (4  a;  H- 7)  (4  a; -5).  9.  {ah -\- c)  {ah -{- d). 

6.    (5  +  m)(5  +  »0-  10-    (2a-7  6)(2a  +  8  6). 

WRITTEN   EXERCISES 

Find  the  products : 

1.  93  .  95.  5.  993  •  985. 

2.  197.191.  6.  (_3aj  +  ll)(-3a;-6). 

3.  (a;  +  14)  (a;  -  19).  7.  (15a;  +  23)  (15-r*;-25). 

4.  (22;-3a)(2^  +  5a).  8.  {4.w +  a){4.w -2  a). 

147.   Type  VIL     Expressions  of  the  form  {ax -{- b){cx -\- d) 
have  the  product  acx^  +  {be  -f-  afl^)  x  -\-bd. 

EXAMPLES 


(.3a  +  5)(4a  +  7)=12a2+(20  +  21)a  +  35 
=  12  a2  + 41^  +  35. 


(ax  —  h)  (ex  —  (?)  =  acx^  —  (6c  +  ad)x  +  6tZ. 

The  coefficient  of  the  middle  term  is  the  sum  of  the  products  indicated 
by  the  curved  lines  above  the  given  expression. 

Actual  multiplication  is  quite  as  simple,  but  practice  in  forming  products 
as  above  indicated  is  a  good  preparation  for  factoring  expressions  of  this 
type. 

ORAL   EXERCISES 

Find  the  products : 

1.  (2a  +  5)(3a-l).  5.    {2x-a){x-h). 

2.  (a-5)(4a4-2).  6.    {Sxy -l){^xy ^-2). 

3.  (3a  +  l)(a-3).  7.    (a;"  -  2)  (3  x"  -  4). 

4.  .(a;  +  4)(3a;-5).  8.    {2x-^  -  a){x^ +  h). 


TYPE  PRODUCTS 


107 


6.  {2xP  —  ab){5xP-{-bc). 

7.  (ma;"  —  2/^)  (px'' +  2  2/^). 

8.  {aby'-c){aby  —  d). 

9.  (cdajp  —  2/")  (acx^  —  chf). 
10.  (x?/"  —  mn)  {zy""  -\-pq)> 


WRITTEN    EXERCISES 

Multiply : 

1.  {ay-hS){by-{-5). 

2.  (2a2/  +  4)(3a?/-5). 

3.  (3  ax  +  6)  (ax  —  c). 

4.  (156  +  2c)(56-10c). 

5.  (^0b'c-l){5b'c-\-d). 

148.   Type  VIII.     (jr  +/)'  =  Jr^  -h  3  jr>  +  3  jr/  +/• 
Since  by  actual  multiplication 

tnerefore,  {x^ay=(  )^  +  S(  )2a  +  3(  )a^ +{  y. 

Also,  (a  +  &2)3=(  )3  +  3(  )62  +  3(  )  (52)2  4.(52)8 

=  (  )  +  3(  )  +  3(  )  +  (  ). 
Since  by  actual  multiplication 

(X  -y)^  =  x^-Sx^y  +  Sxy^  -  y^, 


therefore, 
Similarly, 


(m-n)3=(  )B_3(  )+3(  )-(  )3. 
(a&  -  c)3  =  (a&)3  -  3(a6)--^c  +  3  abc^  -  c^ 
=  (  )-3(  )  +  3a6c2-c3. 


Similarly,  (a  -  6  +  c)3  =  (a  -  &)3  +  3(a  -  b)^c  +  3(a~b)c^  +  c^ 
=  (  )+3(  )+3(  )  +  (  ). 


WRITTEN    EXERCISES 
Expand  by  Type  VIII : 


1. 

(m  +  nf. 

9. 

(2a- 3  6)^ 

17. 

{x^-y^f. 

2. 

ip-qf. 

10. 

a{a-\-bf. 

18. 

(i^'-i2/)^. 

3. 

{a-xf. 

11. 

ax{x  —  yf. 

19. 

{x^-y-)\ 

4. 

(2  a  +  xf. 
{a'-\-b'y. 

12. 
13. 

(a6  +  cdf. 
{x  +  lf. 

20. 
21. 

{a'b  +  aby. 

5. 

{m-{-n-py. 

6. 

(^^2  _  ^2y^ 

14. 

{2>x-lf. 

22. 

(m  +  71  —  p)K 

7. 

{ai-2by. 

15. 

{f-l)\ 

23. 

{a  +  b-^cf. 

8. 

(a-  Scf, 

16. 

(cr  +  6-)^ 

24. 

{2a-\-b+cy. 

108  A  HIGH   SCHOOL  ALGEBRA 

REVIEW 

ORAL  EXERCISES 

State  the  products  of : 

1.  x(x-i-S).  4.   3a(3&-4c).         7.   (3  +  0(^-0- 

2.  x(a  +  b).  5.    b(a  —  c).  8.   (a  —  y)(a-\-y). 

3.  4r(r-2).  6.  p(m-p).  9.    (7- a;)  (7 +  a;). 

State  the  square  as  indicated : 

10.  (a  +  xy.  11.    (1-xy.  12.    (T-yy. 

WRITTEN   EXERCISES 

Write  the  products : 

1.  (2a-l)^  6.  (20  +  1)  (20-1). 

2.  (10 +  3)^  7.  (8a  +  x)(Sa-x). 

3.  (ox-y)(5x-{-y).  8.  (3a- 5)(4a-6). 

4.  (ax-\-S)(ax-S).  9.  (6  a  - 12)  (7  a  +  15). 

5.  (ax^-l)(ax^-S).  10.  (2 a -3  6)  (7a +  5  6). 

Eemove  the  parentheses  and  unite  terms  where  possible  : 

11.  (3 a; -1)2+ 2(4 a; +  3)2.  13.    (79y+(92y.' 

12.  5  (7  2/ -4) -(4  2/ +  3)2.  14.    (ab  -  cy -h  (ab  +  cy. 

15.  (x-3y)(x-{-Sy)  +  (x-5yy. 

16.  3a;(7?/-4)-(2i»  +  32/)2. 

17.  133  .  127  or  (130  +  3)  (130  -  3). 

18.  4a(5-l)  +  2(3a-&)2. 

Find  the  product : 

19.  (l-x){l-]-x)(l  +  x')(l-]-x')(l  +  a^)(l-\-x^^)(l-^-a^'). 

20.  Show  by  multiplying  that 

s (s  —  a) (6  +  c) 4  a (s  —  b)(s  —  c)—2 bcs 
is  identical  with 

s(s  —  b)(a-\-c)—  b(a  —  s)  (s  —  c)  —  2  acs. 


CHAPTER   XII 
FACTORING 

149.  The  factoring  of  algebraic  expressions  depends  largely 
upon  a  knowledge  of  products,  and  for  this  reason  the  special 
products  most  used  in  factoring  were  brought  together  and 
emphasized  in  the  previous  chapter. 

I.  ROOTS 

150.  Preparatory. 

1.  (4-2)(+2)=?  (_2)(-2)=? 

2.  State  a  number  which  taken  twice  as  a  factor  produces  4. 
State  another  number  which  taken  twice  as  a  factor  produces  4. 

3.  According  to  Exercise  2,  how  many  square  roots  has  4  ? 
What  are  they  ? 

4.  Similarly,  name  the  square  roots  of  9 ;  16  ;  25  ;  36. 

151.  Signs  of  Square  Roots.  Every  number  has  two  square 
roots  which  differ  only  in  their  signs. 

Thus,     Vi  =  +  2  or  -  2  ;  because  ( +  2)  ( +  2)  =  4,  and 
_  (_2)(-2)=4. 

Va^  =+  a  or  —  a;  because  (+«)(+«)  =  a^,  and 
l-a){-a)=  a2. 

It  should  be  noticed  that,  although  either  square  root  taken  twice  as 
a  factor  produces  the  given  number,  the  product  of  the  two  square  roots 
is  not  equal  to  the  given  number. 

152.  The  sign  ±  is  used  to  denote  that  a  number  may  be 
taken  either  positively  or  negatively. 

Thus,    Vi  =  +  2  or  -  2     is  v^^ritten       Vi  =  i  2. 
Also,  Va^  =:  +  a  or  —  a    is  written     Va^  =  ±  a. . 

The  positive  square  root  of  a  number  is  called  the  principal 
square  root. 

109 


110 


A  HIGH   SCHOOL  ALGEBRA 


ORAL  EXERCISES 

State  the  two  square  roots  of  each  number : 


1. 

25. 

5. 

225. 

9. 

144. 

13. 

49  a  V. 

2. 

49. 

6. 

81. 

10. 

36. 

14. 

25i)V- 

3. 

121. 

7. 

625. 

11. 

a?h\ 

15. 

\gH\ 

4. 

196. 

8. 

169. 

12. 

36a26l 

16. 

^^imrn^ 

153.   The  square  roots  of  a  monomial  may  often  be  found 
by  factoring. 

EXAMPLE 

Find  the  square  root  of  576  m V. 

By  trial,  576  =  2  •  2  •  2  •  2  •  2  •  2  •  3  •  3  =  23  •  2^  .  3  .  3, 
and  m^n^  =  mn^  •  wn^. 

.-.   V576m^=  ±  23  .  3  .  m  •  w2 
=  ±  24  mw2. 


WRITTEN   EXERCISES 


Find  by  factoring : 


1.    V625 


2.  Vll02oa;«. 

3.  V256  a'b\ 


4.  V225a^?/6. 

5.  V3025  xy. 

6.  V961  a'b\ 


7.    V3136mV. 


8.  Vl69(a  +  6)2. 

9.  V9216  {x-yy. 


154.    Cube  Root.     A  cube  root  of  a  given  number  is  a  num- 
ber whose  third  power  (or  cube)  equals  the  given  number. 

For  example : 

4  is  a  cube  root  of  64  because  4  •  4  •  4  =  64. 

be  is  a  cube  root  of  b^c^  because  he  ■  be  -  be  —  b^e^. 

—  3  a&  is  a  cube  root  of  —  27  a^h^. 
2(a  +  &)  is  a  cube  root  of  8(a  +  by. 

—  a{b  +  c)  is  a  cube  root  of  —  a^{h  +  c)3. 

The  sign  of  the  cube  root  is  the  same  as  the  sign  of  the  power. 


155.   Evolution, 
evolution. 


The  finding  of  roots  of  numbers  is  called 


FACTORING  111 

ORAL  EXERCISES 

Name  a  cube  root  of : 

1.  a\                    5.    27a^fm\  9.  125{p-qy. 

2.  27  a\               6.    S{x-i-yy.  10.  ofiy\h  -  c)\ 

3.  -So?h\          7.    -a?{h  +  c)\  11.  (a  +  6)  ^  (a;  +  2/)^ 

4.  So^fz^.            8.   8m'(m  +  n)l  12.  (a;  —  ?/) ^ (a; -f- 2//. 

II.     MONOMIAL  FACTORS 

156.  When  every  term  of  a  polynomial  contains  a  common 
factor,  that  factor  may  usually  be  found  by  inspection. 
(Sec.  140.) 

For  example  : 

3  a&  is  a  factor  of  3  ahx  —  6  ahy  —  9  ahz^  for  3  ahx  —  6  ahy  r-  9  ahz  = 
3  a6  •  X  —  3  a&  •  2  y  —  3  a&  •  3  ^. 

ORAL  EXERCISES 

State  the  monomial  factor  of  each  expression : 

1.  ah  +  ac-\-ad.  15.    i^  a^hh^ —  2  a^hc  —  2  ahc. 

2.  ab-\-hc-\-h.  16.   3  a/  + 6  ^y  -  9  ay. 

3.  2ax  +  2ay-\-2az.  17.   ic^-6a^  +  12a;. 

4.  im? -\- m^y  +  mh.  18.   x^y'^ -\- xy"^ -\- xyz. 

5.  3  ma?  —  6  m?/  —  9  mz.  19.   3  a.*^?/  —  6  ar^/^  +  ^  ^V- 

6.  5  a6  + 10  a'ft^- 5  a6c.  20.    4  a^a;  -  8  a^  -  6  a^ft*''. 

7.  2a(a;-2/)+2aa;2/.  21.    3  a^6^  -  3  a^  -  15  a^ftl 

8.  cv'h''-^alf  +  ba^h.  22.    1 5  a^a;  -  10  a^^y  _|_  5  ^2^. 

9.  10"+^-^10"+^  +  10^  23.      a2n+453_^3n+4^7_ 

10.  7  aca:2  _  3  5ca:3  _  2  cV.  24.  4i)a;2  -  20^2^^  -  lOpa;^/. 

11.  12  am2  4- 4  ar2  -  6  aVm.  25.  36  5^ -j_  108  g^  -  18i9g^ 

12.  Q>x''y  +  10  a;/  -  18  xY-  26.  5«+i  -  5«+5  +  3  •  5«. 

13.  2aa.-3-f  12aV-24a6a;2.  27.  3a;(22/-l)-6a;2?/-9a?2/^. 

■     5a        2a2         10 a^*  '    2p2         jp3    "^    5^5  * 


112  A   HIGH   SCHOOL  ALGEBRA 

WRITTEN   EXERCISES 
1-28.   Write  the  other  factor  in  Exercises  1-28  above. 

III.    POLYNOMIAL  FACTORS 

157.    An  expression  may  have  a  binomial  or   other   poly- 
nomial factor  that  can  readily  be  found  by  inspection. 
For  example : 

1.  a  +  &  is  a  factor  of  (a  +  b^x  +  (a  +  b)y. 

2.  And  X  +  y  —  z  isa  factor  oi  (x -\- y  —  z)ah  —  3(x  -\- y  —  z)cd. 

3.  3^-5x^  +  Sx-6  =  x2(3cc-5)+  l(Sx  -  6)  =  {Sx  -  b)(x^  +  1). 

Note  that  Sx  —  6  divides  each  term  of  the  middle,  or  grouped,  expression 
giving  the  quotients  x'^  and  +  1. 

ORAL  EXERCISES 

State  a  factor  of : 

1.  {a-\-l)x  —  {a-{-l)y.         4.    {m-{-n-\-p)abi-(m-\-n-\-p)cd. 

2.  {a-{-x)x  —  {a  +  x)y.         5.    (x  +  yy-{x-{-y). 

3.  a{b  +  c)x''-a(b-\-c)y\     6.    (a  + 1^ +  (a -f- l)2-j-(a4-l). 

Supply  the  blanks : 

7.  o.x-\-ay-\-bx-{-by  =  {   ){x-hy)  +  {   ){x  +  y) 

=  [(   )  +  (    )](^  +  2/). 

8.  ax  +  bx  —  ay  —  by  =  {   )  (a  +  6)  —  (   )  (a  -f  &) 

=  [(    )-(    )](«+&). 

9.  2ax^-4.ax  +  ^x-Q  =  {   ){x-2)^{   )(^-2) 

=  [(   )  +  (   )](^-2). 
10.    6a  +  3&  +  9c  +  2ax-h6x  +  3caj  =  (    )(2a  +  64-3c) 

+  (   )(2a+6  +  3c)  =  [(   )  +  (   )](2a  +  6  +  3c). 

WRITTEN   EXERCISES 

Factor : 

1.  a^  +  flj^-Vaj  +  l.  b.  a^  +  ab  —  ac  —  bc. 

2.  :ii?  —  2y  —a?y  +  2x.  6.  ax-\-x  —  ay  —  y. 

3.  ax  —  ay  +  bx  —  by.  7.  x  —  a+ix-af.. 

4.  ax-{-3a-}-bx-\-3b.  8.  5  h' -  4.h' -\-10  h-S. 


FACTORING  113 

9.  6m^H-4m^  — 9  m  — 6.  13.  4:  (;ix -\- bx  —  4:  ay  —  by. 

10.  xy  —  by  —  b -\- X.  14.  4a^  +  a-  — 4  a  — 1. 

11.  x(z-Gy-y(a-z).    ■  15.  a^-4a;2  +  2ic-8. 

12.  a{x-{-iy-\-3x-\-3.     '  16.  x^-x-2x-\-2. 

17.  (a -{- b -\- cy -\- ax -\- bx -\- ex. 

18.  (m  +  r)V-  H-  my  -\-  ry  —  m  —  r. 

19.  x{p  —  y)  +  ap  —  ay. 

20.  aoj  +  a?/ —  £c(a;  +  2/) 

IV.     SQUARES   OF  BINOMIALS 

158.  Since  {x  ±  yy  =  x'^  ±2  xy-\-  y"^,  a  trinomial  is  the  square 
of  a  binomial,  if  one  term  is  twice  the  product  of  the  square 
roots  of  the  other  two,  but  not  otherwise.     (Sees.  141,  142.) 

For  example  : 

a2  +  i4a  +  49  =  a24-2.7a  +  72=(a  +  7)2.      _ 
Here  14  a  is  twice  the  product  of  Va^  and  V49. 
25  w2  -  30  m  +  9  =  (5  m)2  -  2  •  3  .  5  w  +  32  =  (5  m  -  3)2. 
16  a6  -  8  a3  +  1  =  (4  a^y  _  2  .  4  a^  .  i  +  (i)2  =  (4  ^3  _  i)2. 

Test  by  squaring.  f 

WRITTEN  EXERCISES 

Factor : 

1.  x"^ -}- 2  ax -\- a\  12.    9  x^*  —  12  cc"  +  4. 

2.  a;2-2ma;  +  m2.  13.    x'y' -2 xy^ -^1. 

3.  4,x^-4x-{-l.  14.   ^a^-^a  +  i. 

4.  9x2-12aj4-4.  15.   lx'-\-^xy  +  iyK 

6.    xy-^2xy-^l.  16.   pY-lM  +  i. 

o         1 
6.    a2x^2s,  _j_  2  a^^^/c  +  c2.  17.    1— — +  -=- 


7.    a^b"^  +  2  a?>mn  +  mW. 


18.  (l-i))2-6(l-i))  +  9. 

19.  a;2-2(a-6)H-(a-6)l 

9.  p2^4.pq-^4.q\  20.  4(a2  +  l)^  +  4(a2  + 1)  +  L 

10.  4  wV  —  4  joa;  4- 1'.  '  A 

21.  -,  +  20  +  25a^. 

11.  25  a'b'- 10  abc-^c\  af 


114  A  HIGH  SCHOOL  ALGEBRA 

22.  2/^«  — 4  2/"4-4.  30.  aj2'»+4a;"»+4+2(.'y'''*+2)+JL 

23.  aV-a^aj  +  J.  3^^  p^xY- 6  pqxy -\-9. 
,4.  (.4-.)^  +  2(.  +  ,)  +  l.  3^^   a._,,u-2Aa^, 
^^'  ^^"^^■^^2*  33.>«'^4-49-14p3n. 

26     l-|-?4-l.  ^^-   49a;2  4-8l2/2-126a;y. 

X     x-^'  12 

27.  (m-?i)2-2(m-n)a;4-a;'.  ^^'   m^'^m^'^'^' 

28.  (a  +  5)^-2(a  +  %4-2/'. 

29.  9(/>  +  l)2-6(p  +  l)-M.  •   2/2-^    ■^0^2- 

37.  x''-^2xy-\-y^  +  2(x  +  y')z  +  z\ 

38.  a'  +  2a(/)-g)+p2_2pg-}-g2. 

REVIEW  EXERCISES 

Factor : 

1.  ax  +  ab-\-ay\  4.    (x -\- yy -{- a(x -{- y). 

2.  l-2?/Hr2/2.  5.    16  m^  -  24  mn  +  9  n^. 

3.  Sa^x  —  6ay-\-9abxy.  6.  px-\-py -j-qx-^-qy. 

7.  i  +  ^  +  l.       8.^-2  +  2-:.      9.  i-^+1. 

.t2     £c  ^2  p2  y^     y^ 

10.  ax  —  bx  —  a  4-  b.  ,  ^  ,  ,  ,         , 

'  19.    mxy -\- nxy -j- pxy -\~  mcd 

11.  (m  — 2)2  — 3  m +  6.  +wcd+j9cd 

12.  Wa'-Sa'-i-l.  20.   4  a;+6  2/+8;s-h2  aaj+3  a^/ 

13.  ap-\-bp—aq  —  bq.  +  4 as;. 

14.  4a2-64--4a-62-  +  l.  21.  p2_2p^  +  g2_^2(i) -g)r 
,  ^  6  x^      9a;  ,  12a;2  -\-r\ 

15. -f • 

y         y"       ^f  22.   2/2p-42/^  +  4-2(2/P-2) 

16.    81  ic2  _^  121 2/2  -  198  a;?/.  4-1. 

1  7        /y.n+1  _i_  /v.n+2          m^n 

1/.    a:      +«      -X.  23.    abx-\-aby-abz-'^Gdx 

18.    5  a2'*+ V  - 15  a^'^-ifep+i.  -  3  cd?/ -f- 3  cc?;^. 


FACTORING 


115 


V.     THE  DIFFERENCE  OF  TWO  SQUARES 

159.    The  factors  of  the  difference  of  two  squares  are  the 
sum  and  the  difference  of  the  numbers  whose  squd.res  are  given. 

For  example  : 

The  factors  of  a^ft^  —  1  are  ab  -\-  1,  ab  —  1,  because 

{ab  +  l){ab-l)  =  a^b-^-l. 
The  factors  of  a*  —  4  c^d'^  are  a^  +  2  cd,  a^  —  2  cd,  because 

(a2  4- 2  cd)  (a2  _  2  cc?)  =  a*  -  4  c2d2. 

ORAL  EXERCISES 
Read  and  supply  the  blanks  : 

1.  t'^-v''  =  (t-v){         ). 

2.  a''^-4:b'y=(a'-2by)(        ). 

3.  4.x'-y^  =  (2x-tyX         )• 

4.  9x^-4:y^  =  {3x-2y)(         ). 

5.  a V^  —  ^z^'*  =  [ao;"^  —  (     )][aa.-"»+(     )]. 

6.  25s2_49i2^[5s  +  (     )][5s-(     )]. 


WRITTEN    EXERCISES 

Write  the  factors  of : 


1. 

l-y\ 

7. 

^x^-y\ 

13. 

262  -  242. 

2. 

Slu^-6'iv\ 

8. 

a'-9b\ 

14. 

412  _  312, 

3. 

121^-4:. 

9. 

x''-y\ 

15. 

322  _  282. 

4. 

1  - 144  (f. 

10. 

y'-h 

16. 

a'-36b\ 

5. 

a;y-25. 

11. 

92^-1. 

17. 

7632-6632. 

6. 

lUaV)^-4:9c\ 

12. 

36aj4_49 

f- 

18. 

36x21/2-169^2^2^ 

19.    Calculate  the  area  of  the  shaded  portion  of  this  square,  if 

S 


Side 

(1) 

(2) 

(3) 

(4) 

(5) 

(6) 

s  = 

49 
45 

290 
280 

597 
497 

73 

27 

6a 
4.a 

24  g 
14  g 

116                        A 

Factor : 

20. 

a2_52a;2. 

21. 

x^-y\ 

22. 

4  ct^c  —  5^0. 

23. 

9ie2_l62/2. 

24. 

25a«62_i, 

25. 

25aH'-4.¥f. 

26. 

49^2p+6_l^ 

27. 

49a2-4  62. 

28. 

1  -  a2&2c2. 

29. 

1  - 121  xy. 

42. 

16- 

p2«+4 

4a;2 

2522 

43. 

9  2/2 

49 

44. 

16 

1. 

A   HIGH  SCHOOL   ALGEBRA 


30.  1-x^.  ^^    a^_^^ 

31.  laW-lcW.  b^ 

32.  IGxY  —  '^mhi^  ^_q 

33.  ax^-ay\  '   y^"" 

34.  a?^  —  4  y^. 

35.  c^ic^  —  4  c^. 

36.  81a;y-9. 

37.  225a464-l. 

38.  25  a;2?/4— 36  a^y. 

39.  100a266_25a662^     "      625  a^^ 

160.    The  terms  of  the  given  square  may  be  polynomials,  but 
the  method  of  factoring  is  the  same.  ^ 
For  example :  (  ^^^-t  C       ' 

(«  _  5)2  _  (^,  4.  cy=i(a  -  h)  +  (&  +  c)]  [(a  _  &)  -  (6  4-  c)] 
=  (a  +  c)(a-2&-c). 

(a2  +  5)2  _  (a;2  +  ?/)2  =  (a2  +  5  +  2c2  _^.  y)  («2  +  5  _  a;2  _  y) . 

WRITTEN  EXERCISES 
Factor : 

1.  (x  +  yy-(x-yy,  6.  (a=^  _  1)2  _  (52  _  1)2. 

2.  (a +  5)2- (a -6)2.  7.  ^2  ^  2a6  +  ^2  _  ^2^ 

3.  (p  +  g)2  -  (m  +  n)2.  8.  x^-2xy  +  y-  -  ;^2. 

4.  (a-\-b  +  cy-z\  9.  4a2-4a  +  l-96l 

5.  (a  -  2  6)2  -  (3  6  +  c)2.  10.  16  0^62  _  a2c2  _  6  ac  -  9. 

11.    p2^2  _  IQpf.  4.  25  -^2  ^  10^^  _  25^2^ 

12.  a^- 4a6 +452  _  9  2/2.  19.  0;-^  +  2 a^^  +  1  -  a262. 

13.  4  a^  +  8  a6  +  4&2  _  1.  20.  (m  -  2w)2  -  36  7/. 

14.  4.-x''-2xy-y^  21.  (2 a? +  ?/)-- (3  x  -  4)2. 

15.  4ic2_4a2_4«5_52^  22.  (3  a -46)2- (5a  +  66)2.    . 

16.  iK2-2a;?/  +  t/2-16c''^.  23.  (2aj  +  2/-2)2- (5a;  + 62/)^ 

17.  (3a;-2?/)2-(2a;+3y)2.  24.  9a2- -  6a-6 +  62- 9  c2. 

18.  {x""  —  2/")2  —  (a;"  +  y^f.  25.  (5  2;"*  +  2  p  —  1)-  —  x^^'y^p. 


FACTORING  117 

y 

S       VI.     FORMING   THE   DIFFERENCE    OF   TWO    SQUARES 

161.  It  is  often  possible  to  factor  an  expression  by  first 
making  it  the  difference  of  two  squares. 

If  the  given  expression  can  be  made  the  square  of  a  binomial 
by  adding  a  square,  it  can  be  made  the  difference  of  two  squares 
by  adding  and  subtracting  the  same  square. 

For  example  : 

«4  +  a%-^  -}-  64  =  a*  +  2  a^"^  +  ?)*  _  a^^^  =  (a^  +  62)2  _  ^252 
=  (a2  +  6'  +  a6)(a2-f  62-  ah). 
a;4  +  4  =  a;4  +  4  a;2  +  4  _  4  ^2  ^  (^2  _|.  2)2  _  (2  x)2 
=  (x2  +  2  -  2  X)  (a;2  +  2  +  2  a;). 

16  a;4  -  X2  +  1  =:  16  a;4  +  8  ^2  +  1  _  9  a;2  :=  (4  a;2  4.  1)2  _  9  ^2 

=  (4x2  +  1)2-  (.3a;)2=  (4x2  +  1  +  3x)(4  x2  +  1  -  3x) 
Test.  16  -  1  +  1  =  16  =  (4  +  1  +  3)  (4  +  1  -  3). 

WRITTEN   EXERCISES 

Express  as  a  difference  of  two  squares  and  factor : 

1.  a^  +  4.  11.    x^-ir^f. 

2.  0^2/^  +  4.  12.    81  p*  + 9^24-1. 

3.  a^  +  U.  13.    a;^  +  25  a;y  +  625  2/^. 

4.  64  +  &4.  14.   16a^64_j_4^2^2^1^ 

6.  0^  +  4.  10«".  15.  81  a^  +  225  ^262  4- 625  6^. 

6.  .'c^«  +  2a;2"  +  9.  16.  625  a^  +  400  o^y  +  256  ?/*. 

7.  a^-6«22/^  +  y.  17.  a'^  +  2a262  +  9  61 

8.  x^  +  3xY  +  ^y^'  18.  0^4  —  8  i»2?/2  +  4  2/1 

9.  16a4  +  4a2  4-l.  19.  4.  a^  -  U  a'b^  +  ^  b\ 
10.  4a;Y  +  3a;y  +  l.  20.  .'c^  4. 2  a^  -  15  2/^^. 

REVIEW  EXERCISES 
Factor : 

1.  3a.-3-9a;2  +  a^_3.  5.   25  a2-40  a6+16  6^-9  c^. 

2.  771'*  —  m^  —  5  m  —  5.  6.    81  a;^  _  ^^2  _  4  ^/s;  —  4  z^. 

3.  16  a;^  +  4  a;2  _p  1_  7^    i^  (y  —  zf-{-5y  —  5  z. 


118  •       A  HIGH   SCHOOL  ALGEBRA 

9.  64  +  2/4.  16.  64r4-jL. 

10.  (m  -  ?i)2  -  4  icy .  17.  36a;2"»+6_l. 

11.  p^  +  4:Z^\  xY       V^ 

12.  (a;«-6)2  +  2a;«-2  6  +  l.  '  f       ^V 

13.  {x^-yf^-1{x^y)z^z\  19-  o^^'' +  81  -  18  a^. 

14.  bx^-^x^  —  hy'^  —  y'^.  20.  x^^y"^^ -\- 2  afy^  +  1. 

15.  (2  a;  -  2/)"+^  -  (2  x  - .?/)".  21.  36  ?7i2  4. 49  p^  _  84  mp. 

22.    15  a^a;?/  +  25  a^x^y  -  10  a^ar^/^ 
3^_^      4^ 
5        15"^  75   * 
24.   4  a;  +  3  2/  —  2;  —  12  aa;  —  9  ai/  4-  3  az. 

VII.     TYPE  x^+px-\-q 

162.  (aj4-«)(«  +  ^)  =  ar^  +  (a4-&)aj  +  a6.  Hence  if  a  tri- 
nomial is  the  product  of  two  factors  like  x-\-  a  and  x  -f  6,^th3 
sum  of  a  and  h  is  the  coefficient  of  the  middle  term,  and  their 
product,  ah,  is  the  third  term. 

The  factors  of  such  a  trinomial  are  seen  at  once  if  a  and  h 
can  be  found  by  inspection. 

For  example : 

In  x2  +  5  ic  +  6,    5  =  2  +  3,  and  6  =  2  .  3. 

...  a:2  +  5 X  +  6  =  a;'-2  +  (2  +  3) a:  +  2  .  3  =  (x  +  2)(a:  +  3). 

Inx2-7x  +  6,   6  =  (-6)(-l),  and  _7=-6  +  (-l). 

...  a;2- 7x4-6  =  x2  +  (- 6-1), x+(-6)(-l)  =  (x-6)(x-  1). 

Inx2-5x-6,    _6=:6(-l),  and  5  =  6 +  (-1). 

.-.  x2  +  5x  -  6  =  x2  +  (0  -  l)x  +  6  (-  1)  =  (x  +  6)(x  -  1). 

ORAL  EXERCISES 
Factor : 

1.  a2  +  3a  +  2.  6.  x^  +  5  a: -f  4.  11.  a;2  +  8a;4  7. 

2.  d}-3d  +  2.  7.  y^-6y-^5.  12.  a;^- 8  a; +  7. 

3.  a2  — 5a4-4.  8.  m^  +  6  m  +  5.  13.  a^  —  a  —  2. 

4.  ^2  —  8 ;2 -f  15.  "  9.  'm;^  +  4w-21.  14.  m^— 7m  +  12. 

5.  m^-m- 12.  10.  2:2-2^-15.  15.  w'^-4:W-21, 


FACTORING  119 

163.  Although  the  inspection  method  of  Section  162  should 
be  generally  used,  it  may  be  helpful  in  some  cases  to  write  the 
various  pairs  of  factors  of  the  third  term  and  then  compare 
their  sum  with  the  coefficient  of  the  middle  term. 

For  example : 

In  a;2  _  17  aj  +  72,  the  pairs  of  factors  of  +  72  are 

72        36        24        18        12        9 
_1        _2        _3        J        J)        8 

and  the  same  pairs  taken  negatively. 

Since  the  sum  of  the  factors  is  negative,  only  the  negative  pairs  need 
be  examined,  and  by  trial  the  pair  —  8,  —  9  is  found  to  have  the  sum  —  17. 

.-.  x2  -  17  a;  +  72  =  (X  -  8)  {x  -  9). 

Likewise,  in  x^  —  x  —  56,  the  factors  of  —  56  are 
_  28         -  14        -  8 
2        4        7 

or  the  same  numbers  with  the  signs  changed  ;  but  since  the  coefficient  of 
X  is  negative,  only  those  pairs  need  be  examined  in  which  the  negative 
number  is  the  larger. 

By  inspection,  —  8,  7  are  found  to  have  the  sum  —  1. 

.-.  x'2-x-b6  =  (x-  SXx  +  7). 

Test  by  multiplication. 

WRITTEN    EXERCISES 

Factor  (use  Section  163  for  7,  10,  12,  16) : 

19.  x'^-\-6x-^5. 

20.  a.2  _|.  9  aj  _|_  20. 

21.  a;2  -  8  aj -h  15. 

22.  a;2  -f  8  a;  4-  7. 

23.  aj2~10xH-9. 

24.  a?2  +  7  a; -h  12. 

25.  x^-  5x  —  14. 

26.  x^  +  2x  —  15. 

27.  s2-|s-i. 


1. 

x'-x-  30. 

10.   x''-7x  -  18. 

2. 

a;2  +  £c  -  30. 

11.    x"^  -{-7  X  —  18. 

3. 

x'-x-  20. 

12.    x''  +  17x-\-60. 

4. 

x''-\-x-  20. 

13.    m2  +  llm  +  28, 

5. 

a;2  _  3  a;  _  18. 

14.    ??i2  -3m-  28. 

6. 

a;2+3a:_18. 

15.    m2  +  3  m  -  28. 

7. 

a2  -1-  a  -  42. 

16.    iy2  +  6?/-40. 

8. 

a2  _  a  -  12. 

17.   2/' -6  2/ -40. 

9. 

m2  _  1  m  + 1. 

18.    f^-^t-^i. 

28. 

{a  +  by -10  (a 
9 

+  6) +  9.         S9.    (m-\ 

120  A   HIGH   SCHOOL   ALGEBRA 

VIII.     TYPE  mx"^  +  px-^q 

164.  The  type  mx^  +  px  -\-  q  may  be  factored  by  a  compari- 
son of  factors  similar  to  that  in  Section  163.  This  trial  method 
is  commonly  called  the  method  of  ci^oss  products. 

EXAMPLE 
Factor:  12  x\- 7  x  -  10. 

From  the  factors  of  the  first  and  third  terms  without  taking  into  ac- 
count the  middle  term,  the  possible  factors  are 

12ic-10  2a;-5  3x  +  2 

x+    1  6x  +  2  4a;  -5 

in  which  the  two  coefiBcients  in  any  column  may  be  interchanged. 

To  find  which  are  the  actual  factors,  it  is  only  necessary  to  multiply 
the  coefficients  and  observe  what  combination  produces  the  coefficient  of 
the  second  term  of  the  original  trinomial,  in  this  case  —  7. 

Thus,         12-10  2-5  3  +  2 

X  X  X 

1+    1  6+2  4-6 


—  2  (not  —  7)  —  26  (not  -  7)  —  7  (correct) 

Therefore,  the  factors  are  3  a;  +  2  and  4  x  —  6.     Test  as  usual. 

It  is  seldom  necessary  to  try  all  the  sets  of  factors  in  their  different 
-combinations.  Simple  conditions  will  eliminate  them  at  once.  For  ex- 
ample, 12  a;  —  10  could  not  be  a  factor,  because,  if  it  were,  the  factor  2 
which  it  contains  would  be  a  factor  of  the  given  trinomial,  and  it  is  not. 
For  the  same  reason,  6  x  +  2,  in  the  second  set,  could  not  be  a  factor. 

WRITTEN   EXERCISES 

Factor : 

1.  6  a^- 25  a; +  24.  8.  10  y^ -\- IS  y  -  S. 

2.  4.x^-27x-7.  9.  21  .^  +  46  «  -  7. 

3.  12  x»  -fll  a;  -  5.  10.  6  x^  _|.  47  ^^  +  35. 

4.  3j9« -f  16p  -  35.  11.  8|)2-30j9  +  7. 

5.  SSp^-Slp-W.  '                 12.  20  2/2 -49 2/ +  30. 

6.  24a;»  +  73a;  +  24.  13.  18  a^  -  19  a;  +  5. 

7.  7y+48  2/-7.  14.  24  2«  -  103  2  +  55. 


FACTORING  121 

15.  3  7^2  +  20  m -h  32.  24.  2ox^-5x-56. 

16.  21  .v'- 61 2/ +28.  25.  15  m^- 38  m +  23. 

17.  26p'x'-10px-S6.  26.  21  ft^- 375- 28. 

18.  15m2  +  8m-23.  27.  25  c' -^75  c  + 56. 

19.  15  22-17:^-42.  28.  22  x^  +  19  a?  -  21. 

20.  22a^-19a;-21.  29.  25z^  +  5z-56. 

21.  26i)2x2  +  62px  +  36.  30.  21  ?/' +  37  y  -  28. 

22.  3p2_20p  +  32.  31.  3m2-4m-32. 

23.  15a2  +  17a-42.  32.  22  aj- +  47  cc  +  21. 

165.  The  methods  for  factoring  given  in  Sections  162,  163, 
and  164  are  of  limited  value,  for  they  determine  the  factors 
only  in  favorable  instances.  Thus,  a^  —  17  cc  +  72  was  readily 
factored  in  Section  163,  but  rc^  —  16  a;  +  72  could  not  be  so 
factored.  A  general  method  will  be  given  in  Chapter  XXX 
by  means  of  which  all  such  trinomials  can  readily  be  factored. 

REVIEW  EXERCISES 
Eactor : 

1.  a^-49  2/'.  3.    aW-bc^\  5.    121  a^p-y\ 

2.  200  -  2  x^  4.    243-3cci2.  6.    a«  -  256  6«. 

7.  1-25  (a +  2  6)2.  13.    ¥z  -  2  bh^ -{' bzK 

8.  l-Sl{x^-\-x-iy.  14.    4a3-48a262  4-144aM. 

9.  2a2-4a-2a^  +  2.  15.    x'^y^z'^p  +  7  x'^^yh^^-K 

10.  14:x''-21x'y.  16.   100  a^  -  61  a^  +  9. 

11.  3a-b'-6a''-b'-^15a'-bK        17.    a'-ib-^iy. 

12.  x^-2x*-\-a^.  18.    z^-^z  —  2z\ 

19.  25(a-6)2+20c(a-6)  +  4c2. 

20.  a^-\-¥-{-c''-2ab-2bc-\-2ac. 

21.  a'^-\-b^—x'^  —  y^  +  2ab—2 xy. 

22.  2ah-b''-{-x^-a\ 

23.  16  (2  a;  -  ?/)2  -  (a;  -  3  2/)2. 

24.  4  a2  +  9  62  +  25  c2  -  12  a6  +  20  ac  -  30  be. 


122  A  HIGH   SCHOOL   ALGEBRA 

25.  rt^- 169  610.  37.  7a;2-3a;-4. 

26.  36  c^  +  47  c2c?2  + 16  d^  38.  10-5  a -15  a^. 

27.  a^-20a  +  aK  39.  3t^-{-7t-20. 

28.  x'^p-xp-72.  40.  10  ^'2 -34  s +28. 

29.  ay^- ay -20  a.  41.  6  c^- 5  cf^  -  21  dl 

30.  2a;3_io^2^12a;.  42.  m^^  26  m +  133. 

31.  xY-^xy-SS.  43.  (a-6)2~15(a-6)  +  50. 

32.  (a;2-l)2-2(i«2_i>)_63  44  21  r^  -  18  7's  -  48  s^. 

33.  8a2+2a-15.  45.  xY  +  2xY-63. 

34.  2o  x^  —  4:1  xY +1^  y^'  46.  9  a;2*  _  a;Ay  _  10  2/2. 

35.  9^4-148^2^64.  47.  4  x^- 9  a^^^g  ^_  1^ 

36.  14a;2-39a;  +  10.  48.  (2  a  +  6)2- 16(3  a  -  6)2 

49.  (3  +  6)2-2(3  +  6)(^-l)  +  (aj-l)2. 

50.  4  aj4«+ Yn+4  _  8  a^n+iyZn+3  ^  ^2  a;^+32/5n+6. 

IX.     TYPE  jr^+Z  AND  jr^.y 

166.    We  know  by  multiplying  that 

(x  +  y)(x'^-xy-^y^)  =  0(^+y^, 
and  that  (x  —  y)  {x^  +  a;^  +  y'^)=g?  —  y^. 

Hence,  one  factor  of  the  sum  of  tv:o  cubes  is  the  sum  of  the 
numbers,  and  the  other  is  the  sum  of  the  squares  of  the  numbers 
minus  their  product. 

Also  one  factor  of  the  difference  of  two  cubes  is  the  difference 
betiveen  the  members,  and  the  other  is  the  sum  of  their  squares 
plus  their  product. 

EXAMPLES 

1.   Factor:  27  a;^  +  8  2/^^ 

27  x^  =  (3  xy.- 
Sy^=(2yy. 
.'.  21  x^-\-8y^  =(Sx  +  2y)[(Sxy  -ZX'2y  +(2yyi 
=  {Sx  +  2y)l9x'^- 6xy  +  4y'^). 
Test  by  multiplication. 


FACTORING  123 

2.  Factor:  8  a^^s  _  125  c^. 

8  a^¥  =  (2  ab)^ 
125c3=(5c)3. 
.  •.  8  a%^  -  125  c3  =  (2  a6  -  5  c)  [(2  a6)2  +  2  a&  •  6  c  -r  (5  r)2J 
=  (2  a6  -  5  c)  (4  a262  + 10  abc  +  25  c2) . 

Test  by  letting  a  =  6  =  c  =  1 :  8  -  125  =  -  3  •  39. 

3.  Factor :  8  x^  —  a^ 

8x6_a6=(2x2)3_  (a2)3 

=  (2  a;2  _  a'2)  [(2  X2)2  +  (2  ic2)  (a2)  +  (^2)2] 

=  (2  x2  -  a2) (4  X*  +  2  a2x2  +  a*). 

167.   Types  a^  —  if  and  0^  +  2/^  may  be  used  in  calculation. 

EXAMPLE 

Calculate :  14^  -  13^ 

143  -  133  =  (14  -  13)(142  +  14  .  13  +  132) 
=  142  +  14  .  13  +  132 
=  14(14  +  13)  +  132 
=  14  .  27  +  132 
=  378  +169 
=  547. 

WRITTEN   EXERCISES 

* 

15.  xy*  —  om^y. 

16.  125  a' -b\ 

17.  (a +  6)3-1. 

18.  27a^fz'  +  S. 

19.  aW  +  (a-\-bf. 

20.  Sm^  — 27  p^gr\ 
21.,  10^-h^\ 

22.  (a-{-by-{b-cy.  25.    (c-^  df  +  (2  c- d)\ 

23.  {3a-\-by-{2a--by.  26.    (x^ -{- ly -  {f  +  ly. 

24.  (x-\-yy-{x-yy.  27.    8  a^  + 125  (6  +  c)3. 


<'ac 
1. 

itor: 
a'  -  b\ 

8. 

a^-l. 

2. 

a'-S  b\ 

9. 

x^-y\ 

3. 

8  a^  +  b\ 

10. 

o^^bK 

4. 
5. 
6. 

21  x?-f, 
f^21z\ 
a^"  -f- 1. 

11. 
12. 

64  a^  - 1. 
8a^  +  l. 

7. 

.V'- 

13. 
14. 

27  2/'p  -  1. 

124 


A  HIGH  SCHOOL  ALGEBRA 


Calculate : 

28.    16^-151     29.    193-181     30.   23^-213.     31.    57' -56'. 

32.  It  is  known  that  the  volume  of  a  sphere  is  -  ttt^,  r  being 

o 
the  length  of  the  radius.     Using  2^-  as  an  approximate  value 
of  TT,  calculate  the  number  of  cubic  inches  in  a  spherical  shell 
whose  outer  radius  is  14  in.,  and  inner  radius  13  in. 

Solution.     The  volume  of  the  outer  sphere  is  f  •  -^7^  •  14^,  and  that  of 
the  inner  sphere  is  f  •  ^^-  •  133. 

Hence  the  volume  of  the  shell  in  cubic  inches  is 

f  .  2^2  (143  _  133)  =  4  .  y  .  547  ^  2292.+ 

33.  Find  similarly  the  volumes  of  spherical  shells  if : 


(1) 

(2) 

(3) 

(4) 

Outer  radius    = 
Inner  radius    = 

16 
15 

19 

18 

25 
23 

36 
32 

168.  We  have  seen  how  to  factor  the  difference  of  two 
squares  and  the  difference  of  two  cubes ;  it  will  be  sufficient 
for  present  purposes  to  note  that : 

The  difference  of  any  tivo  even  powers  is  always  divisible  by  the 
difference  of  their  square  roots,  and  the  difference  of  two  odd 
powers  is  divisible  by  the  difference  of  the  numbers. 

Furthermore,  the  sum  of  any  two  odd  powers  is  divisible  by  the 
sum  of  the  7iu7nbers. 


1.   Factor: 


EXAMPLES 

^10    _   yW^ 


a;io  _  yio  =  (x5)2  _  (y^y  —  (^5  _  y5)(x^  +  y^).     Then  x^  —  y^  is  divisible 
by  aj  —  y,  s-iid  x^  +  y^  is  divisible  hj  x  +  y. 

2.  Factor:  x^  —  y^. 

x^  —  y^  =  (x  —  y)  (x*  +  x^y  +  x^y'^  +  xy^  +  y*) 

3.  Factor:  a^  +  32  6^ 

a^  +  32  65  =  («  4.  2  b)  (a*  -  2  a^b  +  4.  a^b^  -  S  ab^ -{■  16  ¥) 


FACTORING  125 


WRITTEN 

EXERCISES 

Factor  into  two  binomials  : 

1.    mhi''-p'q\ 

4.    A.a'1 

2.    4  m2  —  9  ?i2. 

5.    m^- 

3.    16-2/'- 

6.    xY 

Factor  into  three  binomials  : 

7.    a'^  —  bK         8.    16— pi 

9.^  xY  - 

-64 


State  one  factor  of  each  of  these  expressions  and  prove  by 
division  that  it  is  a  factor : 

11.  mV-1.  13.    ar5«  +  ?/3«.  15.    So?W  +  21. 

12.  x'  —  a\  14.   x^  —  32.  16.   x^^  +  1. 

REVIEW 

169.    In  attempting  to  factor  an  expression  not  an  indicated 
root: 

First.     Remove  any  monomial  factor,  as  in  Type  II. 

Second.     If  the  resulting  expression  is  a  binomial,  apply 
Type  y,  or  VI,  or  IX. 

Third.     If   the   resulting   expression  is  a  trinomial,  apply 
Type  IV,  or  VII,  or  VIII. 

Fourth.     If  the  expression  is  not   one  of  the   above  types, 
attempt  to  group  it,  as  in  Type  III. 

WRITTEN    EXERCISES 

1.  3a2-15a6.  8.  a^  +  z\ 

2.  a2  —  x\  9.  aV  _  2  ax"^  +  1. 

3.  //  _  9  q2^  10.  m2 ^6m-]-5. 

4.  3  a^  -  6  a'b.  11.  x^  -(-  1. 

5.  xy^z  —  xyz\  12.  x'^  —  2  ax  —  S  a?. 

6.  16  a^-  9cl  13.  2/^^  +  1. 

7.  a?  —  x^.  14.  ac  —  ad  -^  be  —  bd. 


126 


A   HIGH   SCHOOL   ALGEBRA 


15  2aj3_54. 

16.  -  a^  +  a;2  +  12. 

17.  2x''-\-5bx-12b\ 

18.  3a2  +  12a5-2a-86. 

19.  x^  —  y\ 

20.  4  a2  -  62  4.  6  a  -  3  6. 

21.  o?  -  ^h"^  —  aG  +  2  he, 

22.  25cc4-10a;2i/  +  2/2-9;22 

23.  x^  —  &  x'f  -\-  y\ 

24.  16  a^  +  24  a26  +  9  &2. 

25.  4  a;2  _  4  ^2/  +  2/2  —  a;22/2. 

26.  25  a«  +  20  a^  +  4. 

27.  a^  +  6^ 

28.  3^2 -7  0^3/ +  2  2/2. 

29.  3  ic2  -f  6  a;?/  —  24  2/^ 

30.  a.'S  +  3  a;''  -  4. 

31.  3  a^  -  6  a;2  -  9  x. 

32.  3a2  +  6a&-2462. 

33.  a:2"»  —  2  a;"*  —  3. 

34.  a2  -  2  a6  -  ac  +  2  6c. 

35.  m^  —  11  m2n2  +  n\ 

36.  a^  4-  hK 

37.  a;4  -  3  a;22/2  +  2/^ 

38.  8+2;^ 

39.  4  a;^  -f  1. 

40.  x^  —  if. 

41.  a2-62_(ci_5)2. 

42.  27a;2i-3a;-2. 

43.  a;2'»  —  y"^^. 

44.  a(a  +  h)—  c{c-\-h). 

45.  a6  -  3  a  -  2  6  +  6. 


46.  3  a^5  -  6  a;2  _  9  y. 

47.  2a3-3a2_2a^-3 

48.  4  aj"  +  2/*  —  5  a;22/2. 

49.  a^  —  3  a2  _)_  1. 

50.  4.x''-\-llx-  20. 

51.  a?2_4cta;_4  62  4_8a6. 

52.  o?  ^a?  —  a  —  l. 

53.  a;2«  +  2aj«  +  1. 

54.  (a  +  by-{c-dy. 

55.  62  -  a2  +  2  ac  -  c2. 

56.  a2  -  62  -  a  -  6. 

57.  ^  +  ?-^_3. 
(P       d 

58.  16  a;2  _^  iQ  ^y  _  9  ^2^ 

59.  a^  +  a;4  +  ^. 

60.  x^—x'^  —  xj  +  1. 

61.  a;3  -  216. 

62.  a^  +  a262  +  6^ 

63.  a^4-8. 

64.  2a2  +  13a-24. 

65.  a2"-|-6a"  +  9. 

66.  a;4-81. 

67.  ^x^  —  llxy  —2  2/2. 

68.  6  aj2  _  13  r^y  _|_  6  2/2. 

69.  a'2/^  —  xy. 

70.  aj2p  _  2  xP?/  4-  2/^- 

71.  a;"  +  a;22/*  +  y^- 

72.  aj4  — 2a;2^i^ 

73.  a26  —  a2  —  a6  +  a. 

74.  a^  -  6  a2  +  1. 

75.  y?  +  .T^2/^  +  2/*- 

76.  a;2«  —  4  a;«  +  4. 


FACTORING  127 

a2         ^       ^„  93.  a)4  +  2a^  +  2a;2  4-2ic+l. 

77. (w  +  0  . 

4  1 

94.  -4-l  +  a;2. 

78.  1  +  a  —  6  —  af>.  ic^ 

79.  12  0^2 -27  2/2.  gg  2a.'3  +  3a;2-2x-3. 

80.  64 -a«.  gg  4  6^-96. 

81.  3a2-15a4-18.  ^^^  55  o.^  -  a.  -  2. 


98.  6  2/3  -  2/2  +  6  2/  -  1. 

99.  ia^2__4_^2, 

100.  a^'-Sa^  +  l. 

101.  a:^  _  a;2  _  a;  +  1. 


82.  a2  +  a  -  30 

83.  a:^  +  3  a?  +  2. 

84.  16  4-4a2  +  a4. 

85.  a^p  +  2/^''. 

86.  16  x"  -  48  X  +  35. 

87.  a^  +  a^  +  l.  1^2.    a^  +  |  a  +  J. 

88.  2  a2  +  3  a&  -  2  h\  ^^g     i  _  i. 

89.  2  a;2  4-  ^2/  -  3  2/'-  ^' 

90.  6a2  +  10a6-462.  104.  2a}-\-ah~Qh\ 

91.  (x^+i  _  a^-253.  105.  12  a2  _  5  a6  —  3  h^. 

92.  p3  +  3^)2  4-3^  +  1.  106.  a:2r»  _  2  a;"  +  1- 

107.  abx^  +  (a2  +  6^)  0^2/  +  a?>2/^. 

108.  ci^a;  -  a'c  +  o}hy  -  a¥x  -  6^2/  +  c^^. 

109.  (a  +  &)(c2-c^2)_(^2_^2)(c_d). 

110.  a2  +  &2  +  c2  +  2  6c  +  2  cd  +  2  a6. 

111.  o2  4_  ^2  _|_  1  4.  2  6  4-  2  a  +  2  a6  -  d2. 

Factor  by  reference  to  : 

(a  +  5  +  c)2  =  a2  +  62  +  c2  4-  2  a&  +  2  ac  +  2  6c. 

112.  m2  +  ^2  4.  ^2  _j_  2  t7ig  +  2  mr  +  2  qr. 

113.  ■  a2  +  62  4-  aj2  +  2  a6  —  2  aa;  —  2  6a;. 

114.  aj2  .^  ^2  4.  25  4-  2  xy  +  10  a;  4- 10  y. 

115.  9  f  a2  4-  m2  4-  2  am  —  6  a  —  6  m. 

116.  «2  _-_  2  f  y  4-  ^2  _  (3  ^  _j_  9  _  6  y. 

117.  25  a;2  4-  9  2/2  +  4  ^2  +  30  xy  -  20  xz  -  12  yz. 

118.  a;4  -  2  a;2a2  -  2  0^262  +  a'  -\- b' +  2  aW. 


CHAPTER   XIII 
EQUATIONS 

170.  Factoring  is  an  important  process  in  the  solution  of 
equations. 

171.  Preparatory. 

1.  Find  the  values  of  the  trinomial  x^—x—2  when  a?  =  1 ;  2 ; 
0 ;   —  1 ;  —  2 ;  5 ;  for  which  values  of  x  does  it  become  0  ? 

2.  Find  the  value  of  the  binomial  x^  —  2x  when  x  —  \\   —  1 ; 
2 ;   —  2 ;  0 ;  3 ;  for  which  values  of  x  does  it  become  0  ? 

3.  According  to  Exercise  1,  what  are  the  roots  of  the  equation 
a;2-a;-2  =  0? 

4.  According  to  Exercise  2,  what  are  the  roots  of  the  equation 
Q^  -2a;  =  0? 

5.  What  roots  are  common  to  the  two  equations  ? 

172.  Equivalent  Equations.     If  two  equations  have  the  same 
roots,  the  equations  are  said  to  be  equivalent. 

Thus,  4  a;  =  12 

and  5  a;  —  15  =  0 

are  equivalent,  each  having  the  root  3,  and  no  others. 

Also,  a:2  -  25  =  0 

and  4  x2  -  100  =  0 

are  equivalent,  each  having  the  root  5,  —  5,  and  no  others. 

But,  ic2  —  25  =  0 

and  a;2  -  8  X  +  15  =  0 

are  not  equivalent,  for  the  first  has  the  roots  5,  —  5,  while  the  second 
has  the  roots  5  and  3. 

128 


EQUATIONS  129 

WRITTEN   EXERCISES 

Write  the  roots  of  these  equations  and  find  which  exercises 
contain  equivalent  equations : 

1.  Sx-6  =  0,  3.   x^  =  4L,  5.    a?  -  4  =  8, 

2.  aj-3  =  5,  4.   a;2  =  2,  6.   x^  =  9, 

2  a;  =  16.  2^2  =  4.  a; +  3  =  0. 

173.  If  two  equations  have  between  them  all  the  roots  of 
a  third  equation  and  no  others,  the  two  together  are  said  to  be 
equivalent  to  the  third. 

EXAMPLES 

1.  One  of  the  equations  a;  —  5  =  0,  and  x  —  3  =  0,  has  the  root  5,  the 
other  the  root  3.  Between  them,  they  have  the  roots  3  and  5,  which  are 
the  roots  of  x"^  —  Sx+15  =  0.  The  equations  aj  —  3  =  0  and  x  —  5  =  0 
are  together  equivalent  to  x^  —  8  a;  +  15  =  0. 

2.  The  equation  (x—  l)(x  —  2)  =0  asks  :  For  what  values  of  x  does 
the  product  {x  —  \)(x—  T)  have  the  value  zero  ? 

The  product  is  zero,  if  either  factor  is  zero,  and  not  otherwise 
(Sec.  114).  .-.  (x  -  1)  (x  -  2)  =  0,  if  x  -  1  =  0,  or  if  x  -  2  =  0,  and 
not  otherwise. 

Thus,  tlie  solution  of  the  equation  (x  —  1)  (x  —2)  =  0  depends  upon  the 
solution  of  X  —  1  =  0  and  x  —  2  =  0.  The  roots  of  these  being  1  and  2, 
the  roots  of  (x  —  l)(x  —  2)  =0,  are  likewise  1  and  2. 

The  pair  of  equations  x  —  1  =  0,  x— 2=0  is  equivalent  to  the  equa- 
tion (x-l)(x-2)  =0. 

ORAL    EXERCISES 

State  the  equations  of  the  first  degree  that  are  equivalent  of 
each  of  the  following : 

1.  («  -  3)(a;  -  2)  =  0.  6.  x{x-b)=:0. 

2.  (a;  -  5)(aj  -  3)  =  0.  7.  (x  +  7)(a5  +  1)  =  0. 

3.  (aj-3)(x  +  2)=0.  8.  aj(a;  +  3)  =  0. 

4.  (a; -f  5)(aj  +  3)  =  0.  9.  a;(a;-a)=0. 

6.    (a;  -  3)(a;  +  10)  =  0.  10.    (a;  +  8)(a;-ll)  =  0. 


130  A    HIGH   SCHOOL   ALGEBRA 

Factor  the  first  member  of  each  of  the  following  equations 
and  state  the  equations  equivalent  to  each  : 

11.  x2- 3  a; +  2  =  0.  16.  a^^- 4  a;- 5  =  0. 

12.  a;2  -  5  a;  +  4  =  0.  17.  a;^  _  9  ^  q. 

13.  a;2-6a;+5  =  0.  18.  a;^  -  2  a;  =  0. 

14.  a:2  +  6a;  +  5  =  0.  19.  3a;2-2a;  =  0. 

15.  a;2  +  8a;  +  7  =  0.  20.  x^- 14  a;  +  33  =  0. 

WRITTEN    EXERCISES 

1-10.    Solve  the  equation  in  each  exercise  above  from  11-20 
by  solving  the  equivalent  equations. 

Find  the  roots  of  each  equation  by  factoring  the  left  mem- 
ber and  solving  the  equivalent  equations  : 

11.  a;2-a;-20  =  0.  28.  a;^  -  20  a;  +  100  =  0. 

12.  i»2  _|_  aj  _  30  =  0.  29.  a:2  -  8  a;  +  15  =  0. 

13.  a;2-3a;-18  =  0.  30.  a;^  _^  9  a;  +  20  =  0. 

14.  .^2  _  ^  _  30  ^  0^  31.  a;2  +  8  a;  +  7  =  0. 

15.  a;2  4-3a;-18=0.  32.  a;^  +  2  a;  -  15  =  0. 

16.  aj2  -  a;  -  42  =  0.  33.  a;^  -  a;  -  6  =  0. 

17.  x'^-Tx-18  =  0.  34.  a;2  _  5  a;  -  14  =  0. 

18.  a;2  -  17  a;  4-  72  =  0.  35.  a;^  +  a;  -  110  =  0. 

19.  0^2  -  a;  -  56  =  0.  36.  a!2  -  5  a;  -  24  =  0. 

20.  a;2  +  7  x  -  18  =  0.  37.  _  a;^  +  a;  +  12  =  0. 

21.  a;2  +  11  a;  4-  28  =  0.  38.  3  a;^  -  7  a;  +  2  =  0. 

22.  a;2  -  3  a;  -  28  =  0.  39.  4  2^  -|_  12  2;  -f  9  =  0. 

23.  a;2  +  6  a;  -  40  =  0.  40.  25  a;^  4-  20  a;  +  4  =  0. 

24.  a;2  4-lla;+24  =  0.  41.  a;^  -  a; -2  =  0. 

25.  a;2  -  3  a;  -  28  =  0.  42.  aj^  -  64  =  0. 

26.  a;2-6a;-40  =  0.  43.  1  -  a^^  ^  0. 

27.  a;2+18a;  +  81=0.  44.  2  3/^  +  12  2/ +  10  =  0. 


EQUATIONS  131 

174.  Quadratic  Equations.  Equations  of  the  second  degree 
are  called  quadratic  equations. 

For  example,  x'^  =  16,  a:^  _  3 ^j  ::=  0,  and  x^  —  5x  +  6  =  0  are  quadratic 
equations  with  one  unknown,  x. 

175.  The  way  in  which  quadratic  equations  occur  in  prob- 
lems is  illustrated  in  the  following  exercises. 

WRITTEN   EXERCISES 

Translate  each  statement  into  an  equation: 

1.  The  product  of  a  certain  number  and  the  number  increased 
by  3  is  70. 

2.  The  product  of  two  consecutive  integers  is  132. 

3.  The  area  of  a  rectangle  whose  length  is  three  times  its 
height  is  75  sq.  in. 

4.  In  the  case  of  a  body  falling  from  rest,  the  distance  d 
fallen  in  the  time  t  is  one  half  the  product  of  a  fixed  number 
g  (the  constant  of  gravity)  and  the  square  of  the  time. 

176.  General  Form.      The  general  form  of  the  quadratic 

equation  is  o      ,  /x 

aoir  -\-ox-\-c  =  0, 

in  which  a,  b,  c,  are  any  known  numbers,  except  that  a  may 
not  be  zero. 


For  exam 

pie: 

a 

h 

c 

3  x2  -  cc  +  5  =  0, 

3 

-1 

5 

a^2-H7a:  =  0, 

1 

7 

0 

4  x2  -  12  =  0, 

4 

0 

-12 

x^  =  0, 

1 

0 

0 

177.  It  is  often  necessary  to  simplify  equations  apparently 
involving  x^  to  see  whether  or  not  a  is  zero ;  that  is,  whether  or 
not  the  equations  are  really  quadratics. 

For  example: 

a;2  _  3      r  4-  2 

— — — =     ^      can  more  readily  be  seen  to  be  a  quadratic  equation 

when  reduced  to  2  x^  —  10  x  —  21  =  0. 

In  this  form  it  is  apparent  that  a  =  2,  6  =  —  10,  c=  —  2L 


132  A  HIGH  SCHOOL  ALGEBRA 


WRITTEN   EXERCISES 

Eeduce  each  equation  to  the  type  form  and  write  the  value 
of  a ;  of  6 ;  of  c : 

2.    Sx^-5  =  ^~'^ 


2 

2  5     ' 

5       ~7* 


5. 

x  +  -=3. 

X 

6. 

12-.  =  f. 

7. 

x  +  6     x^ 

7          2 

8. 

2x-9     2x' 
3             5 

178.  Incomplete  Quadratic  Equations.  A  quadratic  equa- 
tion which  lacks  either  its  absolute  term  or  its  term  in  x,  that 
is,  in  which  either  c  or  6  is  zero,  is  an  incomplete  quadratic 
equation.  The  two  forms  of  the  incomplete  quadratic  equation 
are 

a 
ax^  -\-hx  =  0. 

It  is  unnecessary  to  consider  the  case  where  6  and  c  are  both  0,  because 
X  would  always  be  0  whatever  the  value  of  a. 

179.  Solution  of  Incomplete  Quadratic  Equations. 

I.    The  incomplete  quadratic  equation  x^  -|-  -  =  0  is  solved  by 

a 

transposing  and  extracting  the  square  root  of  both  members. 


EXAMPLES 

2.   3  x2  =  75.     .-.  x^  =  25  and  ic  =  ±  5- 


3.    ax2=  h.    .'.  x^  =  -  and  a: 
a 


In  Example  3  the  value  of  x  is  the  square  root  of  the  fraction  - ,  and  it 
is  sufficient  to  indicate  this  by  the  use  of  the  radical  sign.  ^ 


EQUATIONS  133 

II.    The  incomplete  quadratic  equation  ax^  +  bx  =  0  is  solved 
by  factoriiig  (Sec.  173). 

Thus,  ic(<:Kc  +  6)=0,  in  which  cc=0  and  aa!;+ 6=0  from  which  x  = 

One  value  of  x  in  this  case  is  always  zero  (Sec.  114),  and  the  other  is 
the  root  of  the  linear  equation  ax  +  h  =  0. 

EXAMPLES 

1.  x2  — a;=0.     .-.  a;(a;  —  1)=  0  and  X  =  0,  x=  1. 

2.  3  a:2  -  10  X  =  0.     .-.  x{^x-  10)  =  0  and  a;  =  0,  x  =  ^. 

WRITTEN   EXERCISES 
Solve  the  following  equations  : 


1. 

x^  =  169. 

14. 

121  0^2^1089. 

2. 

x"   -  121  =  0. 

15. 

7a:2_448  =  o. 

3. 

a^2  _  144  ^  0, 

16. 

i/;2_i|  =  0. 

4. 

a;2_81=:0. 

17. 

2s2-f  =  0. 

6. 

a;2  _  49  =  0. 

18. 

3x2-i|  =  0. 

6. 

x'  -  625  =  0. 

19. 

3^2  +  8=  5a;2  +  8. 

7. 

3a;2-7o  =  0. 

20. 

11  x^-{-x  =  0.     . 

8. 

4  ^2  _  100  =  0. 

21. 

40x2_25a;  =  0. 

9. 

5a^2_500=0. 

22. 

9a;2_17=:4a;2_i2. 

10. 

12  ic2  -  1728  =  0. 

23. 

29aj2_30^10aj2_|_46. 

11. 

s2-i  =  0. 

24. 

40x2-43  =  7-10a;2. 

12. 

t^-^\  =  0. 

25. 

7a;2-5  =  4a)2+7. 

13. 

r2-25  =  0. 

26. 

Z{x'-x)  =  2x-'-^x+l, 

27.  In  the  equation  16  t^  —  256,  t  is  the  number  of  seconds 
required  for  a  body  to  fall  256  ft.     How  many  seconds  is  this  ? 

28.  The  area  inclosed  by  a  circle  is  ttt^.  Using  3.1416 
as  the  approximate  value  of  ir,  find  the  radius  of  a  circle  whose 
area  is  12.5664  sq.  in. 

29.  Find  the  radius  of  a  circle  whose  area  is  3.1416  sq.  ft. 

30.  Find  the  diameter  of  a  circle  whose  area  is  28.2744 
sq.  yd. 


134  A  HIGH   SCHOOL   ALGEBRA 

31.  If  a  stands  for  the  area  of  a  circle  of  radius  r  and  tt  be 
taken  as  ^-^-,  a  =  "^-f-  r\  The  area  of  a  certain  circle  is  ^-f-  sq.  ft. 
Find  the  length  of  the  radius  in  feet. 

32.  Find  the  radius  of  a  circle  whose  area  is  ||  sq.  in. ;  -^^ 
sq.  in. ;  ^^  sq.  yd. 

SUMMARY 

The  following  questions  summarize  the  definitions  and 
processes  treated  in  this  chapter : 

1.  What  are  equivalent  equations?  Sec.  172. 

2.  What  is  a  quadratic  equation?  Sec.  174. 

3.  State  the  general  form  of  quadratic  equations.      Sec.  176. 

4.  What  is  an  incomplete  quadratic  equation  f  Illustrate  its 
two  forms.  Sec.  178. 

5.  How  may  each  kind  of  incomplete  quadratic  equations  be 
solved?  Sec.  179. 

HISTORICAL  NOTE 

We  have  seen  that  the  solution  of  equations  and  the  factoring  of  alge- 
braic expressions  are  closely  related,  and  we  have  found  it  advantageous 
to  employ  factoring  in  solving  equations.  If  Diophantos  had  known  this 
method,  he  would  have  found  two  roots  of  the  quadratic  equation  instead 
of  stopping  with  one.  For  example,  he  showed  in  his  work  Arith- 
metica  that  {x—  V)  (x  —  2)  z=  x^  —  ^  x  -\-  2^  and  could  find  one  root  of 
equations  like  x^  —  Sa:  +  2  =  0,  but  he  did  not  discover  that  this  equation 
is  the  same  as  (x  —  1)  (x  —  2)  =  0,  which  evidently  has  two  roots,  1  and  2. 

It  is  strange  that  the  application  of  factoring  to  solving  equations  was 
so  long  overlooked,  for  it  remained  unknown  long  after  other  more  obscure 
processes  were  discovered.  To  Thomas  Harriot,  an  English  algebraist  of 
the  seventeenth  century,  belongs  the  credit  of  first  reducing  an  equation 
by  factoring.  The  English  were  proud  to  boast  of  Harriot's  knowledge 
of  mathematics,  and  Sir  Walter  Raleigh  sent  him  to  Virginia  to  survey 
the  new  colonial  territory.  He  afterward  returned  to  England  and  made 
other  improvements  in  algebra,  among  them  the  use  of  small  letters  in 
place  of  capital  letters  to  represent  numbers  ;  but  Harriot's  ignorance 
of  negative  immber  prevented  him  from  applying  the  factoring  process 
to  any  but  particular  cases  of  equations. 


CHAPTER   XIV 
FACTORS   AND   MULTIPLES 

180.  Integral  Expressions.  Algebraic  expressions  which  do 
not  contain  fractions  are  called  integral  expressions,  but  this 
classification  is  generally  understood  to  refer  to  the  letters  in- 
volved. 

Thus,  a,  2  ab,  a%,  a  +  x,  {2x-  y)'^,  are  integral  expressions. 

And,  ia,  -,  — ,  ,  -— — ^,  are  fractional  expressions,  although 

a    3a         c  4 

\  a  and  ^~        are  integral  with  respect  to  the  letters  involved. 

Equations  are  called  "  integral "  whenever  the  fractions  involved  are 
confined  to  the  coefficients. 

ORAL  EXERCISES 

Select  the  fractional  expressions  from  the  following : 

1.  a+i&.  Q.  ^~~y^^ 

2.  aj  +  i. 

^  7. 

3.  ^x  +  -- 

c  8. 

a-\-h  +  c 
'  5        *  9. 

5.    i(6  +  c).  ,  ^ 

11.  In  Exercises  1-10,  select  the  expressions  that  are  frac- 
tional with  respect  to  the  coefficients  only. 

12.  From  the  same  Exercises,  select  the  expressions  that  are 
fractional  with  respect  to  the  letters  only. 

13.  Select  all  the  fractional  expressions  after  all  possible 
reductions  have  been  made. 

10  135 


2a 

¥-'■ 

4a;  +  8y 

4 

2a      36 
2c      5d 

136  A   HIGH   SCHOOL   ALGEBRA 

181.  Common  Factor.  An  expression  that  is  a  factor  of  each 
of  two  or  more  expressions  is  called  a  common  factor  of  the 
expressions. 

182.  This  chapter  is  concerned  with  integral  factors  only. 

183.  Algebraic  expressions  which  have  no  common  literal 
factors  are  said  to  be  algebraically  prime  to  each  other. 

184.  Preparatoky. 

1.  What  is  the  degree  of  2a^?  Of  3a'b?  Of  (a-\-xy? 
(See  Sec.  33,  34.) 

2.  What  is  the  degree  of  3  a^b  with  respect  to  a  ?  With  re- 
spect to  6  ?     With  respect  to  ab  ? 

185.  Highest  Common  Factor.  The  highest  common  factor 
(h.  c.  f.)  of  two  or  more  algebraic  expressions  is  the  algebraic 
expression  of  highest  degree  that  is  an  exact  divisor  of  each 
expression,  including  both  the  numerical  and  literal  parts. 

For  example,  to  find  the  h.  c.  f.  of  3  a^b,  —  6  ab'^,  and  9  abc  : 

The  literal  common  factor  of  highest  degree  is  ab.  The  greatest 
common  divisor  (g.  c.  d.)  of  3,  —  6,  and  9  is  3.  Hence  the  h.  c.  f.  of  3  a%, 
—  6  ab'^,  and  9  abc  is  3  ab. 

Although  —  3  is  also  a  common  divisor  of  3,  —  6,  and  9,  it  is  customary 
to  take  the  g.  c.  d.  with  the  positive  sign. 

If  the  given  expressions  are  factored  so  as  to  have  the  h.  c.  f .  as  one 
factor,  the  set  of  second  factors  will  have  no  further  common  factor,  other 
than  unity. 

186.  In  the  case  of  monomials,  the  h.  c.  f.  is  seen  by  inspection. 
Its  coefficient  is  the  g.  c.  d.  of  the  given  numerical  coefficients, 
and  its  literal  part  is  the  product  of  all  the  different  letters,  each 
with  the  lowest  exponent  that  it  has  in  any  of  the  monomials. 

187.  If  expressions  not  monomials  are  given,  they  must  first 
be  factored  if  possible,  after  which  the  h.  c.  f .  can  usually  be  seen. 

1.   Find  the  h.  c.  f.  of  ab^  +  abc,  and  b'^c  +  ftc^: 

Factoring,  ab^  +  abc  =  ab(b  +  c). 

b^c  +  bc^  =  bc{b  +  c). 

The  h.c.f.  is  the  product  of  the  common  factors  b  and  &  +  c,  or  6(&-Ht;). 


FACTORS  AND  MULTIPLES  137 

2.   Find  the  h.  c.  f.  of  (1  -  xf,  x" -1,  x^ -2  x-[- 1. 
Factoring, 

{I  -  xY  ={\  -  X){\  -  X). 

x^-l={x-l){x  +  \). 
x'^-2x-\-\=(x-\){x-r). 
The  h.  c.  f.  is  1  —  aj  or  X  —  1,  because  these  factors  differ  only  in  sign 
and  the  h.  c.  f.  may  have  either  sign. 

188.  The  h.  c.  f .  of  expressions  not  readily  factored  can  be 
found  by  the  long  division  process,  but  no  problems  will  be 
given  in  this  chapter  requiring  the  general  method. 

189.  To  find  the  h.  c.  f. :  Factor  each  expression  into  its  prime 
factors.  TJien  find  the  product  of  all  the  common  prime  factors, 
using  each  the  least  number  of  times  it  occurs  in  any  of  the  given 
expressions. 

WRITTEN    EXERCISES 
Find  the  h.  c.  f .  of : 

1.  Sxf,x''y.  7.  3x\2a^,4:X%x^. 

2.  xy,  x^y,  xf.  8.  Sa^b(^,  Wab^c^,  10  a''b\ 

3.  10  a^  15  a\  5  a.  9.  4  a^b%  8  ab^c%  12  axybc\ 

4.  a;2  _|_  xy,  (x  4-  yf.  10.  10  a^b%  15  a¥c%  20  a^c^. 

5.  3 a2  +  3 a6,  a2 -  &2.  H.  a'^ ^4:,  a^ -2a+2. 

6.  ax*  -  ay\  x^  —  /.  12.  a^  —  1,  a^  +  a  + 1. 

13.  3ai»2^  -2a%  aV,  -3a6a;. 

14.  3a''  +  2a''b-5a¥,2a^b  +  2a¥. 

15.  x"^  —  a^,  x'^  —  2ax-\-a'^. 

16.  aV-8aa;  +  16,  5aa;-20. 

17.  ax  -{-  ay  +  bx-{-  by,  x"^  -\-2xy  -{-  y"^. 

18.  a2  -  ¥x\  a'^-2  abx  +  b^x\ 

19.  x'^-{-2x-\- 1,  x^  +  a;2  +  a; -t- 1. 

20.  6i»2+19a;4-10,3a;2-13a;-10. 

21.  (a  +  6y-l,  a-f(6-l). 

22.  8a^-l,  4a;2-4i»-f  1. 
23  a;2  —  8  a;  +  15,  a;2  —  5  a;. 


138  A  HIGH  SCHOOL   ALGEBRA 

24.  x'^  +  Sx  +  15,  x'^  +  Sx. 

25.  3aj2-f  ic-2,  3aj2_2a;. 

26.  a^  —  a^^a^  —  S  a?x  -f  3  aa;^  —  ^. 

27.  a;2  +  5  a;  -h  6,  a;2  —  4. 

28.  aj2  -  2  a;?/  -  3  2/^  2  a;  -  6  2/. 

29.  a2  -  a  -  6,  a3  _  9  ^2  _,_  27  a  -  27. 

30.  a^-ie,  16-8tt4-a2. 

31.  m^  —  2  mr  +  r^  m^  —  n^. 

32.  a2  +  2  ajp  +  p^^  ah  +  6p. 

33.  d2-9,  d2  +  6d  +  9. 

34.  {a  -  by,  a"  -  b\     35.  a^  +  a;3^  (^  _|_  ^^2^     ^q    ^2  _  ^^^  ^4  _  y^ 

190.  Multiples.     A  product  is  called  a  multiple  of  any  of 

its  factors. 

Thus,  abc  is  a  multiple  of  a,  of  &,  of  c,  of  a6,  of  be,  and  of  ac. 
Also,  X*  is  a  multiple  of  x,  x^,  x^,  and  a;*. 

191.  Common  Multiple.  An  expression  that  is  a  multiple 
of  two  or  more  expressions  is  called  a  common  multiple  of 
these  expressions. 

Thus,  12  x^y^  is  a  common  multiple  of  3  xy  and  6  x^. 

192.  Lowest  Common  Multiple.  The  lowest  common  mul- 
tiple (1.  c.  m.)  of  two  or  more  algebraic  expressions  is  the  alge- 
braic expression  of  lowest  degree  that  is  divisible,  without  a 
remainder,  by  each  of  the  given  expressions,  including  both 
the  numerical  and  literal  parts. 

For  example,  find  the  1.  c.  m.  of  8  a^b,  6  ahc,  and  —  4  ac^: 

The  literal  common  multiple  of  lowest  degree  is  a^bc^.  The  least  com- 
mon multiple  of  8,  6,  and  —  4  is  24. 

Thus,  the  lowest  common  multiple  of  8  a^b,  6  abc,  and  —  4  ac^  is 
24  a%c\ 

Although  —  24  is  a  common  multiple  of  8,  6,  and  —  4,  it  is  customary 
to  take  the  lowest  common  multiple  with  the  positive  sign. 

If  the  least  common  multiple  is  divided  by  each  of  the  given  expres- 
sions, the  quotients  will  have  no  common  divisor,  other  than  unity. 


FACTORS   AND   MULTIPLES  139 

193.  In  the  case  of  monomials,  the  lowest  common  multiple 
is  seen  by  inspection.  Its  numerical  coefficient  is  the  least 
common  multiple  of  the  given  coefficients,  and  its  literal  part 
is  the  product  of  all  the  different  letters,  each  with  the  high- 
est exponent  that  it  has  in  any  of  the  given  expressions. 

194.  If  expressions  not  monomials  are  given,  they  must  first 
be  factored  if  possible,  after  which  the  factors  of  the  lowest 
common  multiple  may  be  seen. 

1.   Find  the  1.  c.  m.  of  ax,  ac  +  ab,  and  cx^  ■+■  hxK 

1 .  ax  =  ax. 

2.  ac  +  ah  =  a{h  +  c). 

4.    . '.  the  1.  c.  m.  is  the  product  of  a,  x^,  and  6  +  c,  or  ax'^ib  +  c). 

195.  To  find  the  L  c.  m. :  Factor  each  expression  into  its 
prime  factors.  Then  find  the  prodttct  of  all  the  different  prime 
factors,  using  each  the  greatest  number  of  times  it  occurs  in  any 
of  the  given  expressions. 

WRITTEN    EXERCISES 
Find  the  1.  c.  m.  of : 

1.  4  ab,  6  ac.  15.  (a  +  b)  n,  (a  -h  b) r. 

2.  5  a\  10  ax.  16.  3  a%c,  5  aW,  15  a^c. 

3.  6pr,  9pg.  17.  (t  —  u)x,(t  —  u)xyz. 

4.  7  x^,  3  xy.  18.  8  xyz^,  24  x^^,  6  xyH^. 

5.  xyz,  yzw.  19.  ax-^x^,  ax  — x"^. 

6.  abc%  a^b'^c.  20.  a^  +  b\  a"-  -  b\ 

7.  x-\-y,  ax-\-ay.  21.  a  —  b,a-\-b,a'^  —  ¥. 

8.  a''  +  ac,ab-\-bc.  22.  3r  +  2,  9r  +  6. 

9.  bcx  +  bey,  abc.  23.  x"^  —  y"^,  {x  +  yy. 

10.  13  a2  - 13  b\  39  ab.  24.  (a  -b){b-  c),  a"  -  b\ 

11.  ax -{-xy,  abc -{-bey.  25.  2{a—b),2(a^b),a'^-\-b\ 

12.  pq,  apq  —  bpq.  26.  x^  +  y^  x^  —  y^. 

13.  a  (6  —  c),  £c6  —  arc.  27.  1  —  cc,  1  —  x^. 

14.  17  a;2,  51 2/^  17  a^.  28.  x^ -^x-\-2,  x-2. 


140  A   HIGH  SCHOOL  ALGEBRA 

29.  c-\-d,2a  —  Sb.  32.    c^  _|_  ^^  a;^  -  ic?/  -|-  y\ 

30.  x  —  l,x-^l,x^  —  l,  33.    (x—7jy,x'^-\-2xy-^y\ 

31.  a^  -  b\  a^-^ab  +  b\  34.    c(a-  b),  a(b-d). 
35.  (b  —  c)  (c  —  a),  (c  —  a)  (a  —  b),  (a  —  b)(b  —  c). 

REVIEW 

WRITTEN    EXERCISES 

Cancel  the  h.  c.  f .  from  the  numerator  and  denominator  of 
each  fraction : 

^     (a2-52)a;2-2aa;4-l.  3     a;^-lQa;^  +  9 

ax—bx—1  '    (x  —  1) (a:;  —  3) 

5.  By  factoring  the  expressions  find  the  highest  common 
factor  of  x'^  -|-  x'^y^  -f-  y^  and  aj^  +  2/^. 

6.  By  factoring  the  expressions  find  the  lowest   common 
multiple  of  a;2  —  3  x-{-  2,  x"^  —  1,  anu  x"^ -\- 2  x  ■]- 1 . 

7.  By  factoring  the  expressions  find  the  highest  common 
factor  of  (2  a;  -l){a^-  1)  and  (x^  +  x'^  +  x)  {x  -  1)  (x""  - 1). 

8.  By  factoring  the  expressions  find  the  highest  common 
factor  and  the  lowest  common  multiple  of  the  two  expressions  : 

(a;2  - 1)  (a;2  +  5  a;  +  6),  (a;2  +  3  a;)  (a;^  -x-6). 

SUMMARY 

The   following    questions    summarize    the   definitions    and 
processes  treated  in  this  chapter : 

1.  Define  an  integral  expression.  Sec.  180. 

2.  Define  common  factor ;  also  highest  common  factor. 

Sees.  181,  185. 

3.  What  is  meant  by  algebraically  prime?  Sec.  183. 

4.  Define  multiple;  also  common  multiple;  also  loivest  common 
multiple.  Sees.  190-192. 

5.  State  how  to  find  the  h.  c.f.  Sec.  189. 

6.  State  how  to  find  the  I.  c.  m.  Sec.  195. 


CHAPTER   XV 

FRACTIONS 

DEFINITIONS   AND   LAWS 

196.  Meaning  of  Fraction.  In  arithmetic  the  fraction  ^  is 
taken  to  mean  either  4  of  the  12  equal  parts  of  a  unit,  the 
quotient  of  4  ^  12,  or  the  ratio  of  4  to  12. 

All  three  questions : 

What  part  of  12  is  4? 

What  is  the  quotient  o/4  -f- 12  ? 

What  is  the  ratio  of  4  to  12? 

are  answered  by  one  fraction,  y%, 

197.  Fractions  in  Algebra.  In  algebra,  similarly,  the  symbol 
-  stands  for  a  of  the  b  equal  parts  of  a  unit,  or  for  the  quo- 
tient of  a-i-b,  or  for  the  ratio  of  a  to  6 ;  but  it  is  usually 
regarded  as  an  indicated  division. 

Symbols  like  -  and    ^  "*"  '^    are  therefore  usually  read  "a  divided  by 
b  4q  -\-p^ 

6,"  and  "  a  +  a;  divided  by  4  g  +p''^ "  ;  but  for  brevity,  they  may  be  read 

"a  over  6,"  and  "a  +  a;  over  4  q  +  j)^." 

198.  The  dividend  and  the  divisor  of  the  indicated  division 
are  called  the  numerator  and  the  denominator  of  the  fraction ; 
together  they  are  called  the  terms  of  the  fraction. 

199.  A  fraction  is  said  to  be  in  its  lowest  terms  when  its 
numerator  and  denominator  have  no  common  factor. 

200.  The  Sign  of  a  Fraction.  Every  fraction,  taken  as  a 
whole,  has  a  sign  before  it,  expressed  or  understood,  in  addi- 
tion to  the  signs  that  the  numerator  and  the  denominator  may 
contain. 

141 


142  A   HIGH  SCHOOL  ALGEBRA 

201.  The  Law  of  Signs  in  Fractions.  The  sign  of  the  quotient 
is  changed  if  the  sign  of  either  the  divisor  or  the  dividend  is 
changed  (Sec.  77)  ;  hence, 

To  change  the  sign'of  either  the  numerator  or  the  denominator 
is  equivalent  to  changing  the  sign  of  the  fraction. 

2  2  2 

Thus,  if  -  is  changed  to  ,  the  latter  is  the  same  as . 

3  3  3 

2                              2                                                      2 
Similarly,  if  -  is  changed  to  ■ ,  the  latter  is  the  same  as . 

o  —  o  O 

If  the  signs  of  both  numerator  and  denominator  are  changed, 
the  value  of  the  fraction  is  unchanged. 


Thus,  ^if  =  ^,  also 


6  (—  a)  _  —  b  (—  a)  _  a_ 


b     b  d(-  c)      -d{-  c)      c  .  d 

ORAL  EXERCISES 

State  expressions  equal  to  these,  and  having  no  negative 
signs  in  the  numerator  or  denominator  of  the  fractions : 

1. 
5. 


-3 
5 

2. 

7 
-9 

3. 

-4 

-7' 

4.    -^ 

X 

4 

-  d 

9. 

(-2)  (3..) 
(-a)(-b) 

13.     3a(-26)_ 
5e 

-gt 

10. 

-4 

(-3)5- 

11          ^'' 

3 

•  (-3)(-5^) 

1 

11. 

—  m 

(-2)(-3x) 

—  am 

p(-q)      ■ 

■     (-«)(-&)■ 

(-«)(- 

-6) 

19. 

2(-3aj) 

1«      -8» 

7. 

8. 

c  "'     a{-b)  5(-3y 

Express  with  denominator  x  —  y: 

17.  -^L.  19.    JZiL.  21.    ^^^. 

y  —  X  y  —  ^  y  —  ^ 

18.  (-3)(-m)         2Q    (-2a)(-56).     ^^    36(0-6) _ 

y  —  X     '  y  —  ^  y  —  ^ 

Express  with  numerator  m  —  r: 
23.'-=!^.      24.      '•- "^    .      26.1^^.      26.         *— ™ 


2  2a;  +  3«/  -3$  (-5x)(32/) 


FRACTIONS  143 


27.  How  are  the  fractions  ^        ,  ^         related? 

a  —  6     &  —  a 

Compare  similarly : 

__     h  —  a        T  a  ~b       b  —  a  -,  a  —  b 

28.    and  — - —  :    and 


—  2  2         —c—d  c-\-d 


29.    ^^(-^)(-^)  and  ^^;    (-«)(-^)(-4  and  ^^^ 


WRITTEN    EXERCISES 

For  each  of  the  following  write  an  equal  fraction  preceded 
by  the  sign  +  and  having  the  same  denominator  as  the  original 
fraction : 

—  X  _         1  .         3a4-2ft 

1. •  o.    —  •  y.    — 

y  a 

2. —'  6.    -  — .  10.    - 

-y  b 

3.    -1^.  7.    -^^.  11.    - 

3q  2m 

_3m  g     _9lIiA.  12. 


4a;  +  5 

5m-l 

3H-4aj 

2-0^ 

a^-3 

^  +  5 

8  a  +  b  1  —  t 

13.  For  each  of  the  fractions  in  Exercises  1-12  write  an 
equal  fraction  preceded  by  the  sign  +  and  having  the  same 
numerator  as  the  given  fraction. 

202.  If  the  numerator  of  a  fraction  is  of  the  same  degree  as 
the  denominator,  or  of  higher  degree,  the  fraction  may  be  re- 
duced to  an  integral  expression  or  to  an  integral  expression  and 
a  fraction  (mixed  expression). 

To  do  this,  divide  the  numerator  of  the  given  fraction  by  the 
denominator. 


For 

example 

X 

a;2  +  l 
2  +  a;  +  1 

_  1 

X 

-^         X2 

+  x-\-\' 

20x2 

-  5  a;  4-  3  _ 

20x2 

5x         3    _ 
lOx     lOx 

:2X 

-h 

3 

10  X 

10  X 

10  X 

go 

■■ 

1  -'■} 

4X 

■'■  , 

*l^ 

144  A  HIGH   SCHOOL   ALCxEBRA 

WRITTEN    EXERCISES 

Eeduce  to  integral  or  mixed  forms : 

at-\-at\  g    s(p-^uY  ^^    a^-3a 

pu 

„    25  a'b' 


w 

a 

IV 

2M-k 

nl 

k 
aOO  +  a' 

K    100 

-try 

2. 

3.  ----- 

4.  j/lO^V  9. 


5.    V      -'/  .  10 


5  a6 
375^2 

25x'y  ' 
a'-\-b\ 
a-{-b  ' 


a-2 

12. 

36xy-\-5 
9x 

13 

2a«4-3a2  +  l 

.  <^ 

14 

a^  +  ?/  +  3 

a;2 

1  K 

4a;^  +  10a?2_^5 

^2  a?  -  a  2  a;2 

203.  Mixed  expressions  are  changed  to  the  fractional  form 
by  reversing  the  process  of  Section  202.     That  is, 

Multiply  the  integral  part  by  the  denominator  of  the  fraction 
and  add  the  product  to  the  numerator  of  the  fraction.  TJie  sum  is 
the  numerator  of  the  result. 

For  example  : 


WRITTEN    EXERCISES 
Change  each  expression  to  a  fraction : 

1.  a  +  2b-\--'  e.  17x*  +  Sa^-h-'  11.  p'-^. 

d  or                    pv 

2.  x  +  ^'  7.  a2+2a&  +  &2_^-.  12.   1-  —  . 

3  z  b                     X 

3.  a'  +  $\.  8.  ^  +  ^+^.  13.  p  +  ^. 

oa^  x-j-1                      100 

4.  x^-^2xy-{--'  9.  xr-\-2-—^'  14.  vt -\-- - 

y  x^  —  2                       V 

^    ,      5  _-        2      o     V  — 4  -e    o        4:X^—2x—5 

5.  a?  — IH -•  10.  ay^  —  6—^ — -— •  15.  ox 

x-\-l  y^                                  x  —  1 


FRACTIONS  145 

204.  Principle  of  Reduction.  Applying  Section  128,  the  value 
of  a  fraction  is  unchanged  if  both  numerator  and  denominator  are 
multiplied  or  divided  by  the  same  number. 

For  example : 

^=3and^  =  12^3^     ^ 
2  2.2       4 

24^,^^^24_^^8^4^ 
6  6-3      2 

^  =  «^;   also  ^  =  ?. 
c      ac  ay     y 

205.  To  Reduce  Fractions  to  Lowest  Terms,  divide  both 
numerator  and  denominator  by  all  factors  common  to  them,  or 
by  the  h.  c.f  of  the  numerator  and  denominator. 

The  division  may  be  indicated  by  canceling. 

For  example,  ^y^^^  is  reduced  to  ^  ^y  dividing  both  numerator 
and  denominator  by  a  +  cc,  their  only  common  factor. 

206.  Law  of  Exponents.  The  law  of  exponents  in  division 
(Sec.  129)  applies  to  fractions. 

For  example  : 

^  _         a  ■  a         _     ci^     _  JL  .  also  —  =  ^^^   =  A_ 
a^     a  •  a  ■  a  '  a  •  a     a^  ■  a^     a^ '  0*0      a^aH     aH ' 

x^  _  X-  x-x-  X  _  x^  ■  X  _     .     ,      14  a^b^c  _  2  ■  7  a^bcab  _  o    ;, 
x:^~    x-x-x    ~    x^    ~     '  7  a'^bc   ~      7  a^bc      ~ 

^^  means  m  factors,  each  a,  in  the  numerator,  and  r  factors,  each  a, 
«'■ 

in  the  denominator.  They  can  be  canceled  from  both  numerator  and 
denominator,  one  by  one,  until  they  are  exhausted  in  either  the  numer- 
ator or  the  denominator. 

ORAL  EXERCISES 

Eednce  to  lowest  terms,  and  express  without  negative  signs 
in  the  numerator  or  the  denominator  of  the  fraction  itself : 

1.  -.  3.  — ^.  5.    — ^.  7.  ^  . 

a'  a^^  2f  16^ 

0    3  6^  ^    m^  6    —  8     ^^^^ 

'  ^W  '  m''  '   -t''  -15r' 


A  HIGH   SCHOOL  ALGEBRA 

3c3 
-2^' 

11.  _-!''. 

4  s-' 

1^-       Jo.- 

15.    1^\ 
10" 

—  a 

12.  3a^ 
15  a» 

940 

16. 

146 

9. 

10. 


WRITTEN    EXERCISES 

Reduce  to  lowest  terms  and  express  without  using  negative 
signs  in  the  numerator  or  denominator  of  the  fraction : 

—  a^hx                          m^'^'^qv^  m^at 

o>   — — •  y.   . 

2  qm  m" 

m  24  icy  2;^ 

_,    mas             %  a(—  b^)(—c^) 

-6a^bc^               'am'                        '  Sa'b{-c') 

4    ^^^^^                          8    ~  ^'"^^^                12  ^"~V"^' 

•  2*a2p'                         *     a^b^'^    '                  '  x'^'Y'^^' 

207.  Sometimes  the  common  factors  are  made  more  ap- 
parent by  factoring  either  the  numerator  or  the  denominator, 
or  both. 

For  example  : 

ax  —  ay  _a(x  —  y)  _a 
ex—  cy      c(x  —  y)      c  ' 
a'^  —  m^   _  (a  +  w)(ff  —  w)  _  a  ■}- m 
4  (a  —  m)  4  (a  —  m)  4 

-5a;-10a    _-5(a;  +  2a)_     -5     _  5 

ic2  +  4ax  +  4a2        (x  +  2  a)2        x  +  2a~  x  +  2a' 


WRITTEN    EXERCISES 
Reduce  to  lowest  terms : 

a^  —  ¥  ab  —  ac  ^    1  —  y^ 

'    b  —  a'  '36  —  3c'  '  a-{-ay 

„3ic  —  3i/  .      a^  —  ci?  ^    a  —  b 

2.  =^  .  4.  .  6.   . 

7  x  —  7  y  4aH-4ic  6  —  a 


il'-f 

ah  -f-  hp 

0?  +  ax 

n.  '''-y\ 

ah  +  hx 

y  —  X 

aj2-2x'  +  l 

12.         '''-•'       . 

FRACTIONS  147 

ct-t  ^^    a-  +  2  gp  +P^  JL3.    (^^  — a;^)^  ^ 

(a-a;)2  2/* 

14    ^!±1' 

m^  —  2  mr  +  i^ 

208.  When  several  fractions  have  the  same  denominator, 
that  denominator  is  called  their  common  denominator.  The 
common  denominator  must  evidently  be  a  multiple  of  the 
given  denominators. 

209.  When  the  common  denominator  of  several  fractions  is 
the  l.c.m.  of  their  denominators  it  is  called  the  lowest  common 
denominator  (1.  c.  d.)  of  the  given  fractions. 

210.  To  reduce  fractions  to  their  lowest  common  denominator: 
Find  the  I.  c.  m.  of  their  denominators  for  the,  I.  c.  d. 

Divide  the  I.  c.  d.  by  the  denominator  of  each  fraction  and 
multiply  both  terms  of  each  fraction  by  the  corresponding  quotient. 

WRITTEN    EXERCISES 
Change  to  fractions  having  the  1.  c.  d. : 


1. 

2       3        1 
3  a<'  4  ic^'  6  a.*^ 

2. 

ahc 
b'  3c'  2d 

3. 

a          c          b 
2  6x'  abxy^  3  acx 

4. 

X    y    z 
a'  b'  c' 

5. 

x^        2/'       2;2 
2 ah'  Sac'  4.hc 

6. 

a           1 

7. 

c           a 

a  —  c    a-\-c 

8. 

4              2 

ax  -{-x^^  ax—  x^ 

9. 

a-\-x    a  —  x 
a  —  x'  a-\-x 

10. 

4  x"^               xy 

3{a  +  hy  6{a''-b^) 

11 

X'                    ?/2 

a2  +  62'    a2_52 

12. 

1           1          ahc 

l-a;'l-»2  •    a-h'  a  +  b'  a'-h^ 


148  A   HIGH   SCHOOL   ALGEBRA 

3  1  x-1 


13. 
14. 
15. 
16. 


8(1 -a^)'  8(1 +  ic)'  4(1 +  a)') 
111 


4a3(a  +  6)'  4a3(a-6)'  2a\a''-b^) 

1 1 1 

2{x-yy  4.(x^yy  6(x-yy' 

X  y  2 


{^  -y)(y-  ^)   (^  -  2/)(^  -  2;) '  {y  -  z)(x -  z) 


ADDITION  AND   SUBTRACTION   OF  FRACTIONS 

211.  Preparatoky. 

1.  What  is  the  sum  of  |  and  i  ?     Of  i  and  |  ? 

2.  How  must  fractions  be  expressed  before  being  added  or 
subtracted  ? 

3.  What  is  the  sum  of  -  and  -  ?     Of  -^  and  -^? 

b  b  abc  abc 

4.  Subtract  —  from  ^.     Also,  —  from  —  • 

be  be  pq  pq 

212.  To  Add  or  Subtract  Fractions.     1.   Find  the  led.  of  the 
given  fractions.     TJiis  is  the  denominator  of  the  result. 

2.  Reduce  the  given  fractions  to  fractions  having  the  I.  c.  d. 

3.  Find  the  algebraic  sum  of  the  numerators  of  the  fractions 
resulting  from  step  2 .     Tliis  is  the  numerator  of  the  result. 

4.  Reduce  the  results  to  lowest  terms. 


EXAMPLES 


Add  -^  and    ^  ^ 


a  —  x  a-{-x 

1.    The  1.  c.  d.  is  {a  -x){a  +  x)=  a^ 
2. 


a a(a  +  x)  _a^_+ax 


a  —  X       a^  —  x"^       a^  —  x^ 
3      3a    _Sa(a-'X)  _Sa'^ —  Sax 
'   a  +  x~     a^-x^     ~     a2  -  x2 
^      ■       ^       ■     3  a    _  g^  +  ax  .  3  g^  -  3  ffx  _  4  g^  _  2  ax 


a  —  x     a  +  x     a^  —  y? 


FRACTIONS  149 

2.   Add  ^ ^ c 

6(c  — a)(6  — c)'       c{a  —  h){a~cy       a(a—b){c  —  b) 

1.  The  1.  c.  m.  is  abc  (a  —  b){b  —  c){c  —  d). 

Before  dividing  the  1.  c.  m.  by  the  given  denominators  we  notice  that 
a  —  c  in  the  second  denominator  is  the  same  as  c  —  a  in  the  1.  c.  m.  with 
fKe-slgns  changed.  Also  c  —  6  in  the  third  denominator  is  the  same  as 
h  —  c  in  tlie  1.  c,  m.  with  the  signs  changed.  Hence,  the  signs  of  each  of 
these  factors  may  be  changed  and  by  changing  the  signs  before  the  second 
and  third  fractions,  the  denominators  of  the  three  fractions  are  now, 
6(c  — a)(6  — c),  c(a  — 6)(c  — a),  and  a(a  — &)(6— c),  and  the  given 
fractions  are  all  positive. 

2.  Dividing  the  1.  c.  m.  by  each  denominator  in  turn  we  have  ac^a  —  6), 
ah{h  —  c),  and  hc{c  —  a). 

3.  Multiplying  the  results  in  step  2  by  the  numerators  of  the  corre- 
sponding fractions  and  placing  the  sum  over  the  1.  c.  m.  we  have  the 
final  sura  : 

g^c  (g  -  6)  +  a&2  (5  _  c)  +  bc^  (c  -  a) 
abc{a  —  b)(b  —  c){c  — a) 

If  the  numerator  and  denominator  of  the  sum  have  a  common  factor, 
it  should  be  canceled  so  that  the  result  will  be  in  its  lowest  terms. 
Test  :  Let  a  =  l,6  =  2,  c  =  3.     Then  the  given  fractions  become 
1  2  3  1      1_^3^29^ 


2.2(-l)      3(-l)(-2)       1(^1)(+1)  4      3  12 

The  result  becomes    3(- 1)  + 4(- 1)+ 18(+ 2)  ^  29^ 
6(-l)(-l)(+2)  12 

These  being  equal,  we  are  reasonably  sure  that  the  work  is  correct. 
If  in  checking  the  work  in  fractions  with  arbitrary  values,  the  num- 
bers chosen  make  any  denominator  zero,  other  values  must  be  used. 

WRITTEN    EXERCISES 


Add: 

1. 

a 

b 

bx' 

X 

2. 

a 

c 

6c' 

xy 

3. 

m 

V 

n 

Q 

4 

3p 

P 

4g 

'  2q 

5.  -—,  ^^—'  9. 

6.  ^,  ^.  10. 
11. 

8.    -,  -.  12. 


6^'  12z 

a 

b 

X 

y 

1 

5 

x^' 

xy 

b 

w 

a 

V 

\  —  x^  \-\-x 
2't-4.        1 


c    c-f-a; 
a      2  a 

a-\-x  Sx-\-5a 
a  —  x  2(a  — ic) 


160  A   HIGH  SCHOOL   ALGEBRA 


13. 

1       1 

mn    pq 

Subtract  the 

16. 

3    2 
6'  h 

17. 

3a   2h 
b  '   a 

18. 

1     1 

X   3x 

14. 

3 

5 

d-f  1 

fraction 

.  frorr 

19. 

5 

X 

a2 
0^' 

20. 

7 
'   a' 

4 

ab' 

15      ^-P    Ap  +  5 


22. 
23. 


4.x 

2x 

7 

3(a+5) 
5 

1-x' 
8 

2 

21.    -, -.  24. 

213.  Signs  before  Fractions.  Since  the  sign  before  the  frac- 
tion relates  to  the  fraction  as  a  whole,  the  ichole  numerator  is 
added  or  subtracted,  as  the  case  may  be,  the  bar  of  the  fraction 
having  upon  the  numerator  the  effect  of  a  parenthesis. 

For  example  : 

a     d— e_a  —  (d  —  e)_a  —  d  +  e 
c  c  c  G 

a     d  —  e  +  f_a  —  (d-e+f)_a  —  d  +  e-f 
c  c  c  c 

a      .     2  a    _ 6  a  — 7  ax  _ ax—  a  ,2ax  +  2a     6a  — 7  ax 


cc  +  1     a?  -  1         x2  -  1         x^-l         x2  -  1  x2  -  1 

_  ax  —  a  +  (2  ax+2  a)—  (6  a  —  7  ax) 

a;2  —  1 
_  10  ax  —  4  a 

WRITTEN    EXERCISES 

Perform  the  operations  indicated : 

_    x  —  2y     2x  —  y 
^'   ""5~"^~3~* 

6.   _^  +  -i^. 
2  m^g     2  mq^ 

Sa}b     5  ab" 


1. 

ox. 

X     7  X 

2. 

x  —  xy      x  —  xy 
4              2 

3. 

x-[-y     x-2y 
z      '      2z 

4. 

1     3-x 
X         x^ 

5  x^y      3  xy 
ab     be 


8.   ±j^IL-4.xbc. 


9.   ^-2.-Sad  +  ic\ 
c      d 


10. 

5x      lOy 

11. 

2xy            x^ 

12. 

1'+     1     . 

a—b     a+b 

13. 

a  ,  a      a       ^ 

14. 

a       3b        1    . 
6V     aV     a6V 

22. 

a         a  —  b 
2b     2(a  +  &) 

23. 

2y     3(x-y) 

24. 

a2  +  62       52.^^2 

a6            6c 

25. 

a  +  6      64-c     c-\-a 
a            6            c 

rk/« 

a  —  b     c  —  a     b  —  c 

161 


26. 


ab  ac  be 

27.   ^^ 3+      2« 


a  —  a?  (a  +  a?)^ 

15.  -     t  +  !^  +  a'b.  28.    -1        -?^. 
b     a     ab  (x^-\-iy     x^-\-l 

16.  -^^ .^^.  29  1  1  ^ 


a;4_a4     a4_^a;*  2(x-l)     3(a;.+  l)      a;^ 

m         _2^  2x^—y'^     y'^—z^     z^—x^ 

7      ^      3g*  '        x"  2/^  2;2 


18    ^_^-A.  31         ^1^1^ 

'  y     xy     wy  ,  '  2{a-by2{a^by  a^^b'' 

19.  ^  +  ^  +  ^.  32.   2-^  +  ^;. 
bed  x^-\-y^     x^  —  y^ 

20.  a  +  TT-^— +  ir^-  33. -^- -+      ^ 


6a5cc?/     3aca;  (ct  — 6)(6  — c)      a^  — 6^ 

21.   ^iil_^z:l/  34.    -^ ^ 

xh/         xy^  x^  —  y^      (x-\-yy 


35. 36 


1  1 


c(a  —  b)      a{b  —  a).  {a—b)(c  —  a)      (b  —  a)(c—b) 


b  .  c 


3<y      ^ I [_ 

(6  -c)(b-a)      (c-  a)  (c-b)      (a-  b)  (a  -  c) 
38.    , ^ +  , ^ ^  + 


{b-c)(c-a)  (G-a){a-b)  {a-b){b-c) 

39.           «  +  6  I           b-^c          ^          c  +  a 

(6  — c)(c  — a)  (c  — tt)(a  — 6)  (a  — 6)(6  — c) 
11 


152  A  HIGH   SCHOOL  ALGEBRA 


,0.    ,         \        ,  +  ,.       i       .+  ' 


(a-b){a-G)      {b-c){b-a)      (c-a)(c-6> 

41.  a(b-\-c)         ^        b(c-\-a)        ^      c(a-\-b) 

(a  —  b)(c  —  a)      (b  —  c){a  —  b)      (c  —  a)(b  —  c) 
^2  52  ^2 

42.  ——A r  +  7^ TT^ r  + 


(a  —  b){a  —  c)      (b  —  c)(b  —  a)      (c  — «)(c  — &) 


x(x-y)(x-z)      y(:!/-z)(y-x)     z{z-x){z-y) 

214.  Preparatory. 

Eead  as  the  sum  or  difference  of  two  fractions : 

1    ^+y .  2    ^+^-  3    ^-±1.  4    i±±^. 

3     *  *       6  '   x-1  '     a  +  9  * 

215.  Separating  Fractions  into  Parts.  It  is  sometimes  de- 
sirable to  separate  fractions  into  two  or  more  addends,  by 
reversing  the  process  used  in  the  addition  of  fractions. 

For  example : 


1. 

a  +  b _a     b 
c        c     c 

2. 

ax  +  5y_ax5y_x.l 
Way        10  ay      10  ay     10 y     2  a 

3. 

x^  +  x'^  +  l_^^         1 
x  +  1               '  x+1 

4 

(a  +  xy  -  S(a  -  X)  _  (a -{- x)^     3(a-x)  _a  +  x         3 

a^  —  x^                 a^  —  xi^        d^  —  x^       a  —  x     a  +  x 

_     a      1      X             3 
a  —  x     a  —  x     a  +  X 

WRITTEN   EXERCISES 

Write  each  fraction  as  the  algebraic  sum  of  two  or  more 
fractions,  reduce  the  fractions  to  lowest  terms,  and  unite  those 
with  the  same  denominator : 

1  5a;  +  8?/  g     a'^x  +  4  b^y ^ 

10  a     '  '        12  ay      ' 

2  6a'^—  Sb^  .     g  +  a;      a^b~5  dx ^ 

14  ab  y              aby 


FRACTIONS  153 

21o}x     '  '    0^-2  ah -{-b''' 

Sg-]-57i  g     (a-iy-ll(a  +  l) 

•    30(^  +  1)  •  a'-l 

6xy  6xz 

10     7y-3.T     4.mj-9ax  +  15i/      ^    .^ 
3y  Say 


11. 


8  a6  6  a&c 


10  a?  +  5 

a3  _  53  ^  (2  g^  -  7  ab  +  5  6'^) 
3  (a -6) 

14  a^-6^-3(a-6)      7  a^  + 7  a -(a^- 1) 

a  +  6  5(a-hl) 

15  4(l  +  r)  +  3r(l-r)      7  a(l  +  r)- 6  r(l  -  r) 

1  —  r^  a  —  ar^ 


MULTIPLICATION   OF   FRACTIONS 

216.  Multiplying  Fractions.  The  product  of  two  or  more 
fractions  is  the  product  of  their  numerators  divided  by  the  product 
of  their  denominators. 

2  5^2-5^10 

3  *  7      3  .  7      21 ' 


For  example 


q      nq  b      d      4       ibd 


a 

—  b     a  •  —b      —  ab         ab 

c 

d          c  •  d          cd           cd 

Since  every  integer  or  integral  expression  can  be  regarded  as  a  fraction 
with  denominator  1,  the  above  definition  of  product  also  includes  the 
case  where  one  of  the  factors  is  an  integral  expression. 


154  A   HIGH   SCHOOL   ALGEBRA 

217.    In  simplifying  the  results  of  multiplication,-  canceling 
may  be  helpful. 

For  example :  2 

If  necessary,  factor  before  canceling. 
For  example : 

^  .       51      ^    i'^  Z-V(   ^      3a; 

34  '  ax  +  x2     2  •  X  "  X(a  +  ^)     2(a  +  x)  ' 

a;2_4  _9  +  6a  +  a2^(a;-  2)<:aM-^C^-r«)2 
3  +  a  '      5^  +  10  5CaH-0(2-K2) 

^(x-2)(3  +  a) 
5 

Only  common  factors  of  the  whole  numerator  and  denominator  may  be 

K  /v.  2 

canceled.     E.g.  in  — neither  the  5  nor  the  x  nor  the  2  may  be  can- 

5x  +  2 

celed  :  but  in  ^^  ~ — ^  the  5  may  be  canceled .     What  is  wrong  with  this 

5(x  +  2) 

indicated  work,  ^^+^^  ? 

ORAL  EXERCISES 


7.  p'-f-' 
4a 

8     ^  .  ^. 

Multiply  each  of  the  following  in  turn  by  12,  and  reduce  to 
lowest  terms : 


Multiply : 

46     1 
•    3x     5 

-h' 

1 

-^ 

c 
3(^ 

2     t  .t, 
5       4 

'1 

7  * 

6.    6a  . 

4& 

'  5c 

9. 

8 
Sax 

11. 

5x 

ac 

13. 

2m 
6aby 

15. 

8(? 
12  c 

0. 

4 

2  by 

12. 

4?/ 
3« 

14. 

7  ao; 
36 

16. 

5t 
24  s 

Multiply  Exercises  9-16  above  in  turn  by : 
17.   3  a.  18.    2  6.  19.    xy.  20.    4aa;.  21.    S  cy. 


FRACTIONS 


155 


Multiply : 


1. 


2. 


X" 


X' 


a*     a' 
cd      c^d 


WRITTEN   EXERCISES 


3.  1^ .  ^y 

5  a    3a6 


103n+2     io2« 


5. 


7c3     *  4^" 


28      a^" 


2/V'' 


Multiply  and  reduce  to  lowest  terms 


7. 


9. 


10. 


12. 


13. 


7 
13  a 

a 


39^ 
49 

15  a;' 


c6^ 
ad 


3"c 


11.    -TL  .  rjr 


9c 
ad 


ajy^g 


14.     - 


31. 


32. 


33. 


34.    11^  .  ^1^^ 


1 


15. 


16. 


17.    =- 


Pi  .  M. 

— ^^^—  •  a6. 
he 


{x  +  yy 


18.    - 


r 


19.  pV 


20. 


21, 


ah 
p^(f 

d^ah 
a^w'  cd 
4^2 


5  a;?/ 

1 

a^¥xy^    ahx^y 

oi^hh    a^h^c 
ah'^c    xy^'^ 

3mVs^  6  x'^y^z^ 
4  aj?/^^^     5  m2?is 

cd?    ey    a  6 


22. 


xy"- 

mn 
3» 


ajy. 


3  mx. 


23. 


A2. 

a&a; 


a6. 


a^  ,  c^ 
6aj    d 


24.    ^.— 


25. 


2^    3a6    3ac 
a   "~c~"2h 


26.    a.-i.- 
bc    a 


27.   3  6c 


28. 


h  +  c 
2  be  ' 


29. 


30. 


a' 


35.    (a; +  3) 


c" 

a;-2 
a; -3* 


36. 


37. 


4  a;      2(a  +  5) 
a  +  5'     Sa^ 

am  —  hm         2 


c  +  1 
38.    (a^-l) 


6r 


aa; 

a  +  1 


156  A   HIGH  SCHOOL  ALGEBRA 

39.    l-xv^f-4--\.  46. 


2       \x     y_ 


x'^-2x-\-l     Sq^ 
q^  x  —  1 

40     -^a'b'f---^  47     l  +  6a  +  9a^        7t 

4.       \b^      bj  '  21  2  +  6  a 

41.    abc^fl-^X  48.  ^^'  ^-^' 


bj  x^-10x  +  2b       8  d 

42.  ..aVf-+iV  49     9-aV16-8a  +  a^ 


1  fx    y\ 

2  \y     xj 


43.    -fljy ^  .  50. 


a_4     a2  4-6a  +  9 
ra;+14    2a-3     aj+2 


l_5a     (a; +  2)2    6a-9 


44.  f.+iy.-iv      51    ^^  -  ^-^'  -^+y 


yj\       yj  x  —  12a  +  ayl-{-x 

3m2  +  2m      3m2+2m-l 


-  &+»)e-'^}  - 


6m2-ll?7i+3       3m  +  2 


Simplify,  but  do  not  multiply  the  factors  in  the  terms  of  the 
final  result : 


53. 


54. 


55. 


x'  +  2y' 

'■-2xy 

a2  _  2  ab  +  62 

x'  +  4ry' 

c^-d' 

m^ 

+  mn  +  7^2 

m^  —  n^ 

2ad- 

■2  ac  +  ce  —  ed 

Sx'-, 

xy 

2x^-7x-4. 

56. 


58. 


59. 


a;2_a;-12    12  x"" -{- 11  xy  -  5  y^ 

-+- 

ax-}- an  —  am  y      x 

1^  ,  2  ,  1     *a;2-2(m-n)  +  (m-?i)2' 
a'2  ?/2 

g^     106a;  +  56v-2062    g^  - x^ -{- b"" -\- 2  ab  ^ 
a-\-b-\-x  Sax -{-4:  ay  —16  az 

m^-1 gg  4-  ^2  ^  ^2  _|_  2  a.5  4-  2  ac  +  2  &c 

(a+6  +  c)2  +  aaj  +  5a;  +  ca^'  ^>(77i  +  1)2+ 3  m  +  3 

g'  +  4  6^  a/>  +  2a6c-3a6c^-a6d 

c  +  2c2-3c^-c(i'  a2  +  262-2a6 


FRACTIONS 


157 


218.  Powers  of  Fractions.  Any  power  of  a  fraction  equals 
that  poiuer  of  the  numerator  divided  by  the  same  power  of  the 
denominator. 


For  example 


2\2_  (-^2}^ 


V     3)  (3)-^        9 

\h)   ~b'  b'  b~  b^' 

^V  =  ^.«...  to  w  factors. 
bj       b    b 

a  ■  a"-to  n  factors 


3  a%2  Y 
-5  be) 


I  m  +  r  Y  _  C 
\m  —  rl       ( 


6  •  6  •••  to  n  factors 

(3  a^x'^Y  _     27  a^x^ 
(-5&c)3 

m  +  ry 

ry- 


-  125  &3c3 

m'^  +  2  mr  4-  r^ 
m2  -  2  mr  +  r2 


27a^ 

125  &3c3  ' 


Reversing  the  order, 

_a2j-6_a_+9_  ^  (a  +  SY 
x2-10x  +  25      (a; -5)2 


/a+_3\2 


WRITTEN    EXERCISES 

Write  without  parentheses : 


1. 


2. 


3. 


-  2  a  3  52 


5  a;?/ 


1  +  ^ 


1  + 


13. 


14. 


15. 


2/; 

a_b\\ 

X     yj 

—  abxV 

c^dy  J 

ab 


5. 


6. 


7. 


('-^.)('* 


3  aM^ 


Y. 


4  m^Ti  / 
2a;-3\3 


ab 


a  +  b 


16. 


17. 


18. 


9. 


10. 


11. 


(r- 

f2ax_\\ 


(-Hf^j-  "■( 


2_a^Y 
'S^^yj' 


+ 


1  1^2 

a     6 


/g  +  ^cV 
\a-\-b 


158         A  HIGH  SCHOOL  ALGEBRA 


19.    (-  +  -Y-  23.    [-— 1)  •  27. 


X      yj 


20.    (i,  +  lY.  24.    (l-V^.  28.    r?  +  l 


jn?       J  \x     yJ  \h 

21.    ('?  +  ^Y-  26.    ^--^V-  29.    (1  + 


M 


22.    (1+-).  26.    (  —  -1].  30.    i--c 


1_ 

X 

.J. 

1_ 

■-T- 

X 

2/y 

'x 

y\\ 

J 

x) 

'1 

-lY, 

J 

i-xY 


^  +  .VY.  25.    f^-^y.  29.    (l 

y     x)  \y      xj  \2 


a  J 


a 


Write  each  of  the  following  as  a  power  of  a  fraction : 
31.    ^^2^+1.  35.  ^'^' 


32. 


a^  +  2a6  +  ^'  .       36     100  i>' -  20  p^  4- 1 

ic^  -  2  a:^^^  -f  a.V*  *      g^  -  20  g2  4.  loO  * 


33      a;^  +  4  a;  +  4  ^  3^       a^  _9  ^2  _,_27  ^-27 


4  ar^  +  4  a;  +1  27  a.-^  -  27  x^  +  9  a; 

«'^'     7::^ — 2 T^ — : .  ,,  •  00. 


9a2_l2a6  +  462  a;^  +  2  a^?/ +  a? Y 

DIVISION   OF   FRACTIONS 

219.  Reciprocal.     If  the  product  of  two  numbers  is  1,  each 
is  called  the  reciprocal  of  the  other. 

Thus, 

5  and  ^  are  reciprocals  of  each  other,  because  5-^  =  1. 

-  and  -  are  reciprocals  of  each  other,  because  _.-=!. 
b         a  ha 

220.  Division  of  Fractions.        To  divide  by  a  fraction  multiply 
by  the  reciprocal  of  the  fraction,  that  is,  by  the  fraction  inverted. 

4       2  4       ^ 

Thus,  to  divide  -  by  -    multiply  -  by  - , 
o       o  5       2 

and  to  divide  -  by  —  multiply  -  by  ^ . 
b       ^y  b         X 

Test.    If  the  quotient  is  correct,  the  product  of  the  quotient 
and  the  divisor  equals  the  dividend. 


FRACTIONS  159 

WRITTEN   EXERCISES 


1.  ±H-- 

a 

2.  U 
3. 

4. 

5.    ---. 
b      b 

m  .  a 

7.    — -^ 
8. 


Divide : 

H- 

9. 

.    Sp 

'  Uq 

17. 

3a2      2  a 

2  62  •  36 

a     a 

10. 

n 
m 

.1. 

18. 

5a;2      ^ 

t     y 

1  .  1 

a  '  b' 

11. 

X 

— 1- 

y 

I. 

x' 

19. 

15  m^      5 

3     5 
— J — . 

a     b 

12. 

1     :    ^. 

3a6      ab 

20. 

ab  .   6 
c    '  c2' 

a  .  1 

b  '  b 

13. 

y  ' 

X 

21. 

6  a  .  a 

6    '  X 

111  ,  a 
n   '  b 

14. 

mn 
pq 

m 

22. 

7a2    .     14 
5  6    '  5  a62 

a         c   ^ 
26      3d 

15. 

mx 
ny 

n 

23. 

fz_^yz^ 

XW         X 

m       5m 
5  n     21  n 

16. 

$- 

a 

24. 

m^      10  m 
a;?/^;       3  a; 

25.    A^J^  =  ? 

6a;     ex 

33. 

2x     4a; 
32/  '  32/ 

=  ? 

26.    ^""'^  •  ^^_? 
3  c?/      cy 

34. 

2a;2 
yz 

.  3xyG_^ 
'     zH        ' 

27.        *      :       «'=? 

a;  —  2/      a;  +  2/ 

35. 

Zabx  .  6aV      ^ 
5&    '  10c62      • 

„g      3  6.T   .     aa; 

2a2'^  '  2z^+^ 

—  9 

36. 

2., 
3a; 

a 
~b' 

9a;2      ^ 

42/2      • 

29.    ^^^=9 
6d      cc? 

37. 

a    6 
6  '  c 

c 
e 

"f  f 

30.     ^^    :2^-? 

38. 

a2-62-c2'» 
-     a;2/2; 

,_a^b-c-^^ 

3^      3  a6  .  6  a26 

5  a^c      5  a^c^ 

_  ? 

39. 

2/ 

a;p 

.t-9 

15  cW  '  3cH      ' 

40. 

p2g 

q^r 

>  .  mnp  _  p 

160         A  HIGH  SCHOOL  ALGEBRA 

221.  Since  every  integral  expression  can  be  regarded  as  a 
fraction  with  denominator  1,  the  process  of  division  includes 
also  the  case  where  one  of  the  fractions  is  an  integral  expres- 
sion. 


For  example : 

_3^6=-f-f-f=-f.^=-33,. 

«     :  c=    «     :  ^=    «    .1=     «      or. 
—  b            —b     I— be—  be 

a 
be 

Sec.  201 

by     '  ^          ^        by       -Sa         by 

Sec.  201. 

x^y:  -^-^^V  .-z_     ^y^^ 

WRITTEN    EXERCISES 

Perform  the  divisions  indicated  and  reduce  the  results  to 
lowest  terms : 


3a% 

2  c. 

by 

2a     3 

a26 

62c  • 

c 

-3a2 

>a2. 

bx' 

a^  +  W 

.  a-\-b 

a^-¥ 

a-b 

2ab 

26 

3c2   •  ■ 

—  a^c 

9^2^ 

.   Sxy' 

12  mn^ 

4:m^n 

-bo" 

Sa 

7c52 

21  c^b 

—  Tix^y 

.     70^2/    . 

122 

—  4,zw 

5ab' 

—  5  a^b 

1.    ^l^^-2c.  10. 

by 

2  a     Sa^b  _- 

Z.     rrz — • *  i.±» 

12. 

4.  -    ■   -  ^r^L^.  13. 
a^—  b^     a  —  6 

^     2a6        26 

5.  -:z—r-- —'  14. 

15. 
16. 
17. 

9.        ^r/      ^ ^^^^^-  18. 


15 


25  a2^,  5a6 

a262     a  —  b 
a  —  x     a^  —  x^ 


y        -^y 

af  +  y^    .  x^-\-2xy-\-y^ 
x(x—y)  x^  —  y^ 

2a  +  36  .      c  —  d 

c^cl     '  2a-3b' 

a        a^  —  a 


a-{-x      a-\-x 
4a;2_9y2  ^  2x-\-3y 
a'-W     '     a-b    ' 
4  a;2       .         xy 

S(a  +  6)  ■  6(a2_2,2^)' 

a2  2a6 


-21ccZ       7c2cZ2  a2-62     (a -by 


19. 
20. 
21. 


FRACTIONS                                       161 

a^  —  a?.       a  —  x                «„          x^-1       ,  x  —  1 

a+x    '  —{a  +  xy 
a;2_52  _         5_,_c 

x^-3x-{-2     x-2 
^2     x'^-2xy-Sy^  ,x-Sy 

-he    '  -h{x-\-hy 
2  ax  —  x^     x^-  0?- 

x^-^2xy-\-y^    '    x-\-y 
„^     a'-ay  ,     a'-y'    . 

a{x  -\-a)       x  +  a    ' 

•    Say^^2x-5 

771  — n       —  (a  —  yy 
.  15a^  +  lla;-14 
■  12  0.-2  + 11  a;- 15 

COMPLEX  FRACTIONS 

222.  We  have  so  far  considered  simple  fractions,  those  in 
which  neither  the  numerator  nor  the  denominator  is  in  fractional 
form. 

Since  a  fraction  indicates  division,  the  division  of  two  frac- 
tions may  be  indicated  in  fractional  form. 

I 
Thus,  t  -^  I  niay  be  written  -, 

a 

and  —  -4-  -  may  be  written  _. 

d 

223.  Complex  Fractions.  A  fraction  whose  numerator  or 
denominator,  or  both,  are  fractional  expressions  is  called  a 
complex  fraction. 


For  example  : 

a-\-h 

31 

3a; 

c 
d     ' 

2 
/ 
3 

2+a 

x  +  2' 
3 

4y 
7x 
12  y 

are  complex  fractions. 


224.  A  complex  fraction  may  be  reduced  to  a  simple  fraction 
either  by  performing  the  indicated  divisions,  or  by  multiplying 
both  numerator  and  denominator  by  the  1.  c.  m.  of  their  respec- 
tive denominators.  In  each  case  any  operations  indicated  in 
the  numerator  or  the  denominator  should  be  performed  first, 
as  far  as  possible. 


162  A  HIGH  SCHOOL  ALGEBRA 

EXAMPLES 


x'-y^ 

1.    Simplify: 

a  +  6 
x  +  y 

o}-W 

X2_y2 

a  +  b      x2  -  y2 

.    x  +  y 

^■^y       a  +  b 

'  a^-b^ 

a  +  b       x  +  y  "'^         ^ 


62 


2.    Simplify : 


a       b 
ax     ax 

5  +  c 
by 


^,b_     a+b     axy(a  +  b) 


^ ^  y(a  +  b) 

b  +  c      axyjb  +  c)      x{b  +  c) 


by  ay  ay 

WRITTEN   EXERCISES 

Reduce  to  simple  fractions : 

24£  J_a_  1  1_1 

1.   _1_.  3.   ^.  5.   ^-^  7.   ^ l, 

6a;*  21 6*  a  +  ft'  '  x  —  y* 

12  ab 

2    -  4    ^'-^  6    ^^^^  8    ^-^^^' 

8*  ■    m  +  3  '  *    25  d  '  '     7^  +  1  ' 

a*  a  16  ac  3a6 

ar'-l  1  1 


_5a_^  11.  -^-^ 


15  a^b  ^  —  y 

2  aa; 


a 


2  +  ax  +  l.  12.     «+-^ 


10 ^.  i  + 


a2a^2  —  1  ic     a  —  2  a; 


FRACTIONS 


163 


13. 


14. 


a-3 

3b 

36 
a-3 

1 

1 

Sb      a-3 


■      a       b^ 
b'      a" 

i-l  +  i" 

a2     ab     b^ 


17. 


a_6 

b      a 


ct  .b 
b      a 


a  +  x     a 


15. 


16. 


a^     a 
a     6_2 

T  1^  A 


a  —  x  a-^x 
a-]-x  a  —  x 
a  —  x     a-^x 

a^  —  Sab 
l  +  x 


18. 


c        a 


ca 


_^  +  A  +  A 
be      ca      ab 


a-hh-\-cy      o") 
6  4-  6c  +  ca       J 


19. 


x^-^x^ 


1   2/^ 


ic^- a? 


1- 


REVIEW 
WRITTEN    EXERCISES 
Eeduce  to  an  integral  or  mixed  expression : 


1. 


x'^-hb 


6x^-\-5 
x  +  5 


3. 


Eeduce  each  expression  to  a  fraction  : 
b  ...         1 


a^  —  c^ 


4.    a  -f 


5.   a;-f-l 


X'\-l 


6.   c  + 


62 


a+6 


Eeduce  to  lowest  terms  : 


7. 


x^-\-2xy-\-y^ 


x^-f- 


8. 


9  a;^  -  12  a?  +  4 
9  a;2  -  4 


164  A  HIGH  SCHOOL  ALGEBRA 


a^— 4aH-4  ax -\- bx -\- ex' 

^Q     Sax +  4:  ay  -  16  az  ^^    a^  -  x''+  b^ -\- 2  ab 

■'    10  bx-\- 5  by— 20  bz'  '  a-\-b-{-x 

11    ^-1  14    x^  +  3x^  +  x-\-3 

x^-l  '  2x-^6 

Add: 

17     ci-2b     3b 

c  2c 

1^1  2 


15. 

x      2a; 

X 

12  a 

y 

4* 

18. 

x-2y     2x-y 
Sy           12y   ' 

20. 

19. 

a      b      c 

21. 

l  +  o;      1  —  X      1  —  X 
12  3 


22. 


X  -\-  a     X  -\-  b     X  -\-  c 

5a7  +  4      3  a;- 2     a;^-2a;-17 
x  —  2        x  —  3       aj2  —  5a;  +  6* 


23.      „,^-^    ,   +^+      "^ 


a2  -  a6  +  62     ^  _^  5      a^  +  6' 

24.   (a?  +  y)(^^  +  y^-l)  ,  fa  +  l)(y^  -  0^^  +  1) 
a;?/  y 


I  (a^  +  l)(a?^-y^  +  l) 


a; 


25.    ^A^  +  -^,.  27.    -T-V-T+      -^ 


aj2_2^2     a;3_2/3  a^  +  a^  +  l      a2_^a+l 

26.    -^ ^.  28.  ^-^       +     2^-^ 


a;*-l      a;2  +  l  aj2-7a;-18      a;2-8a;-9 

Multiply : 

29.    ^~^  .  ^_±J:.  32.    a;'^  -  2  a;y  +  a?^ .       1 

X       —  y '  '         ^+  y         x^  —  y^ 

30.  ^^.^y-y\  33. 

3  c?/   a;2  —  xy 

31.         "-^       ."'-'"'■  34. 

(a  -  a!)2     -  a6  -  6     a^  -  6= 


x^y 

x-\-2 

x^- 

■1 

x-1 

x^- 

4 

a-\-b 

ac 

FRACTIONS  166 

35  a^'-9    .  x^-1^  37     -4a6       9  c'^d    ^  -21c 
a:--^  4:x'  x^—'Sx'  '       6  c        —Ta^b     12  abc 

36  4a;2       ^  ^(a^J^  ^g        a-6     ^  a^-4  6^ 

*    3(a  -f  ?>)  *      -xy    '  '    a^  +  2ah'  a^-ab' 


39. 
40. 
41. 
42. 


£c2  _  2/2      x"^  —  4:  xy  -\-  4:  y^' 
4  m  4-  r  10  am  —6m 

25  a2  -  9  *  16  m2  + 8  mr  4- ?'2* 

a;2  4-6a;+  5        .    a^  -25b\ 
a^-{- 10  a25  +  25  aft^  *  x'  +  2a;  +1* 
a;2  4-  6  ga;  +  5  g^  ^   g6  4-  ?^a; 
a;2  +  2  ga;  +  g2     5  gc  4-  ex' 


Divide : 

43.    ^^^ 

b       62 


^^    «c_^g^_^iB2^^  ^g  /g  -  by  .  /g2  -  by 

'    bd  '   be  '  y  '  y^'  '  \x  —  y)    '  Kx^  —  2/7  * 

45.    g4-&  .  (a4-&y^  50^  /m  -  riV  .  fn  -  m\\ 

a  —  b  '     a  —  b  \m  —  p)       v^  —  i>/ 

a;2  +  5a;4-6  .  a;4^5  a;       .    a;  4-  a;^ 

a;2-l        *a;  +  l*  *  1  +  a;  '  (1 4- a;)^* 

47.    -^!^by^^.  52.    ^--16by^^-4a^ 

a;3-l-^a;-l  i)4-^  i^f  +  i^ 

g2-3g-28 ,  g'-7g 

a;* +  4  2/4         ^  a;^  +  2  a;^/ 4- 2  ^/^  * 
g^+lO  0-'^^'^  +  25  a;4  ,     g^  4-  7  g^a;^  4- 10  a;^ 

K  'tZ  ^  1 0  ^2  _  Q  K  ^ 


Simplify : 


5^2  -^      10  a;2  -  35  a^ 


55.    1 i- —  56.    g 5-5 — 

1  d! 

g c 

c  g 


166  A  HIGH   SCHOOL  ALGEBRA 


57. 


58. 

\h{x-2y) 
w?  4-  n^ 

x^-2xy^f 

hx^—l^xy  ' 

SUMMARY 

The  following  questions  summarize  the  definitions  and  pro- 
cesses treated  in  this  chapter  : 

1.  State  three  meanings  of  a  fraction.  Sec.  196. 

2.  Which  meaning  of   a  fraction   is   commonly   used   in 
algebra  ?  Sec.  197. 

3.  When  is  a  fraction  in  its  lowest  terms  f  Sec.  199. 

4.  What  is  meant  by  the  sign  of  a  fraction  ?  Sec.  200. 

5.  State  the  Law  of  Signs  of  fractions.  Sec.  201. 

6.  When  may  the  numerator  be  divided  by  the  denomi- 
nator ?  Sec.  202. 

7.  How  may  a  mixed  expression  be  changed  to  a  fraction  ? 

Sec.  203. 

8.  State  the  principle  that  applies  to   the   reduction  of 
fractions.  Sec.  204. 

9.  How  is  a  fraction  reduced  to  its  lowest  terms  9     Sec.  205. 

10.  What  law  of  exponents  applies  to  fractions  ?     Sec.  206. 

11.  De^ne  common  denominator  ;  oXso  lowest  common  denomi- 
nator. Sees.  208,  209. 

12.  How  may  fractions  be  changed  to  equivalent  fractions 
having  the  lowest  common  denominator  ?  Sec.  210. 

13.  How    are    fractions    added  ?     Multiplied  ?     Divided  ? 

Sees.  212,  216,  220. 

14.  How  is  a  power  of  a  fraction  obtained  ?  Sec.  218. 

15.  What  are  complex  fractions  and  how  are  they  simpli- 
fied? Sees.  223,  224. 


CHAPTER   XVI 
EQUATIONS 
PROPERTIES 

225.  Degree  of  an  Equation.  The  degree  of  an  equation  is 
stated  with  respect  to  its  unknowns.  It  is  the  highest  degree 
to  which  the  unknowns  occur  in  any  term  in  the  equation. 
Unless  otherwise  stated,  all  the  unknowns  are  considered. 

For  example  : 

1.  3  aj  +  1  =  0,   and  4  x  +  7  y  —  3  ^  =  6,   are  equations  of    the    first 

degree. 

2.  4:  y^  —  y  +  S  =  0,  and  x"^  -\-  y"^  —  4:  =  0,  are  equations  of  the  second 

degree. 
S.   x^-l  =  0,x^  +  2y^-4:X  +  S=0,6xyz-{-x'^  =  2y,Sindx^-Sy^  + 
0  =  5  a;  are  equations  of  the  third  degree. 

4.    V  =    ^       is  of  the  first  degree  in  v  and  of  the  third  degree  in  B. 
3 

226.  An  equation  of  the  first  degree  is  called  a  linear 
equation. 

227.  An  equation  of  the  second  degree  is  called  a  quadratic 
equation. 

228.  An  equation  of  the  third  or  higher  degree  is  called  a 
higher  equation. 

It  is  unnecessary  to  define  here  the  degree  of  expressions  containing 
radicals. 

229.  In  order  to  state  the  degree  of  an  equation  its  terms 
must  be  united  as  much  as  possible. 

Thus,  a;2  +  2  X  +  1  =  x^  appears  to  be  a  quadratic  equation. 
But  2  X  +  1  =  0,  to  which  it  reduces,  is  a  linear  equation. 
12  167 


168  A  HIGH   SCHOOL  ALGEBRA 

230.  Terms  not  involving  the  unknowns  are  called  absolute 
terms. 

Thus,  in  x^  -\-  5  =  Sx  —  2  a,  5  and  —2a  are  absolute  terms. 

231.  Fractional  Equations.  An  equation  in  which  the  un- 
known quantity  occurs  in  any  denominator  is  called  a  fractional 
equation. 

2                                                                                         la 
Thus, 3ic  +  l  =  0isa  fractional  equation  ;  also =  6. 

X  x'^       X  +  I 

We  shall  not  define  here  the  degree  of  fractional  equations.  It  will 
be  shown  later  that  the  process  of  clearing  of  fractions  may  change  the 
degree  of  an  equation. 

ORAL  EXERCISES 

State  the  degree  of  each  of  the  following  equations  with 
respect  to  each  unknown: 

1.  a;  +  2  =  0.  9.  ax^  -\-b  =  0. 

2.  ax +  2  =  0.  10.  a:f^  —  by"^  =  1. 

3.  z'-mz  =  4..  11.  a^x^  +  hhf  =  aW. 

4.  x"  -j-xy  =  5.  12.  f  TTi^  =  100. 

5.  ^l^^o.  13.    ^^=6. 

2  4 

6.  ^mv^  =  6.  14.   2  7rr%  =  50.       • 

7.  ^^2^32.  15.    mv^-16  =  0. 

8.  ai  -f-  i  gt^  =  0.  16.   a;/  -  x'^y  =  6. 

Select  the  linear,  the  quadratic,  and  the  higher  equations 
from  the  following : 

17.  a;  +  3  =  0.  24.  ax"^  -\- ax  =z  c. 

18.  x^  =  9.  25.  2  a?  -  8  .T  =  0. 

19.  a^  =  27.  26.  3a;2-ar^-f3a;  =  a^  +  l. 

20.  2  a;  -  6  =  0.  27.  2  a:^  ^  2  x^  -f  a;  +  2  a^. 

21.  ax  =  b  -j-c.  28.  x^  +  5  =  3. 

-     22.    ma^-^px  =  q.  29.    af  —  by'^  ^-1  =  0. 

23.   a^  -  a;2  -  a;  =  3.  30.   5  /  -f-  2  ^  =  6  +  5  y^. 


EQUATIONS  169 

SOLUTION   OF   LINEAR  EQUATIONS 

232.   To  Solve  a  Linear  Equation  with  One  Unknown.     In 

general : 

1.  Clear  it  of  fractions,  if  there  are  any. 

2.  Remove  the  parentheses,  if  there  are  any. 

3.  Transpose  all  the  terms  containing  the  nnknovm  quantity 
to  one  member,  preferably  the  left,  and  all  the  other  terms  to  the 
other  member. 

4.  Unite  the  terms  in  each  member  as  much  as  possible. 

5.  Divide  both  members  by  the  coefficient  of  the  unknown. 


EXAMPLE 

Solve: 

6        '        4       -^"^      2* 

{!) 

Multiplying  each  term  by  12,  the 
1.  c.  m.  of  the  denominators, 

2(2a;-5)  +  3(6x  +  3)  =  60a;-6.35. 

{2) 

Eemoving  the  parentheses. 

4a;- 10 +  18x4- 9  =  60a; -210. 

{3) 

Transposing  -  10,  9,  and  60  x, 

4x  +  18x-60x  =  10-9-  210. 

(4) 

Uniting  terms,  and  multiplying 
both  members  by  —  1, 

38  X  =  209. 

(5) 

Dividing  both  members  by  38, 

x  =  V. 

(^) 

Test. 

2(Y)-5,6(Y)+3_5,,,,     35. 
6                   4            ^^^2, 

WRITTEN   EXERCISES 

Solve  and  test : 

1.  2(x  +  3)-3(a;  +  2)=6(a;  +  5). 

2.  x(x-l)-(2x-l)  =  x(x-{-6). 

3.  2a.-ll±^  =  l^±^. 

2  3 

4.  (x  +  S)^  =  -7  +(5-x)\ 
^     z— 1  ,  z  —  2      n      2  —  3 

2^3  4 

X     4.X-12     9t4-2^q 
'   4"^       5  20 


170  A  HIGH  SCHOOL  ALGEBRA 

^•12         8     ^"^^-—T" 

8.  5(a;  -  4)  +  ^(x  -^)-x{x-l)  =  x(;6-  x). 

9.  1(0^  -  5)  -  i(a;  -  4)  =  1(05  _  3)  -  (a;  -  2). 

2^4     6  ■  3     4     2~  4  ^ 

11  y      3-y_5  +  y      p.,  2  x     5  a?     «^  .  ^.  in 

12  ?_^  =  l4-^_^  iA    ^     a?-8_  X 

'  3     4     2"^5     6*  •^''-  5         4     "20* 

13.   ^  +  ^=6  +  ^.        17.   ^,.5+ 2  (20-05)  =10. 
18.    i{x-S)-^(x-5)=:^(x-15)-\-5. 

233.   A  fraction  that  is  not  given  in  its  lowest  terms  should 
dftrst  be  reduced. 


EXAMPLE 

Solve:  a;(l+6a5)      1^ 

aj2-2aj   ^x  ^^ 

■Simplifying  the  first  fraction,  L+A?  +  -  =  6.  (2) 

X  —  2x 
Multiplying  both  members  of 
(3)  by  the  1.  c.  d., 

x-2  X                 ^          ^  ^  ^ 
Canceling  common  factors  in  (3) 

from  numerators  and  denomi-  x(l  +  Q  x)  +  X  —  2  =  6  x(x  —  2) .  (A) 

nators,  V      '          /    '                           v            /  K'tJ 

Kemoving  parentheses,       "  X  +  6  X^  +  x  —  2  =  6  X^  —  12  iC.  (5) 

Simplifying,  14  X  =  2.  {6) 

.•.X  =  f  (7) 

T--  (^^^  +  r^-^-^^  =6  as  in  Step  (i). 

If  the  given  equation  was  cleared  of  fractions  before  reducing  the  first 
fraction  an  equation  of  higher  degree  would  result  which  would  not  be 
equivalent  (Sec.  172)  to  the  given  equation.  There  would  be  a  factor  x  in 
every  term,  giving  the  value  0,  which  does  not  satisfy  the  given  equation. 


EQUATIONS  171 

WRITTEN    EXERCISES 

Solve  and  test : 

•   a;2_4      ^_2  '    Bx'-Sx 

a;-l  3a;2         2« 

3.   -  =  a; -^'  13 


3             i»-l  x-2     x-\-2     x  +  2 

>.     2/     V'-l  -.^2,2/                ^ 

4.  ^  — -^ =  ?/.  14.   -H ^^ —  = . 

2    2/  +  1  y    y^'-y    yiy-^) 

5.  ^-^  =  -^:zl.  -15.  ^1+1  +  5=-^'-^^. 

cc^     a;              a;2  2/  — 3              {y  —  ^)y 

6.  ^_+-L  =  -A_.  16.    — § ^iL_4.5  =  0. 

a;_l^a;  +  l      a;2-l  2x-S     Zx-\2 

7        2             4^7  ^^      3iK             6            7a;  +  9 


w  — 2      tt34-2      w2_4  ^^5     2  a; +  10     3  a; +  15 

g    3 5^1  ^g    3     2x-^      {x-Z^x  _ 

'   X      x-^^      x^-\-^x  *   4         3ic  4  0.-^      ~^* 

9.   ^±i  = ^ +  _^.  19.   a^  +  2=-J-  +  ^'-^^ 

x  —  \      {\  —  Qi?)x     X  —\  x-\-l      a;+5' 

10    _A^_  =  _i_    ^  20       1       ,      3     ^6^  +  18 
;^(a;  +  2)      a.'-5*  ■   y-3      ?/  +  3       /-9 

^,     ^o  +  l      w  +  2  2w2_3^y_l_5 

<4i. -f- 


22. 


w  —  4      m;  +  7      w^ +  3 10  —  28 
cc  — 2      x-{-2  _      4a;  — 1 
a.' +  3     aj  +  S^aj^  +  Saj  +  lS* 


234.  Decimal  Coefficients.  Equations  in  fractional  form, 
like  those  in  Sees.  231-233  should  be  distinguished  from  equa- 
tions having  decimal  coefficients.  The  former  present  pecul- 
iar difficulties,  as  explained  above,  but  the  latter  yield  to  the 
same  processes  as  equations  with  integral  coefficients. 

If  equations  fractional  in  themselves  contain  decimals,  the 
processes  of  Sees.  232,  233  are  to  be  applied  first. 


172  A   HIGH   SCHOOL  ALGEBRA 

235.  Preparatory. 

We  have  already  used  decimal  coefficients  in  evaluating,  in 
adding,  in  subtracting,  and  in  solving  equations  containing  per 
cents. 

Example:  A  number  less  5  %  of  itself  is  4.75.  What  is  the 
number  ? 

Solution.     1.    x—  .05x  =  4.75. 

2.  .95  X  =  4.75. 

3.  x  =  ^—  =  —  =  5,the  number. 

.95        95 

This  equation  could  be  written  x  —  jf  „  x  =  4^%,  but  this  is  not  so 
simple. 

236.  In  simplifying  equations  with  decimal  coefficients,  it  is 
generally  better  to  work  with  the  decimals  or  midtiply  both  members 
by  a  power  of  10  so  as  to  make  all  the  decimals  whole  numbers, 
instead  of  substituting  for  them  their  common  fractional  equiva- 
lents. 

.      EXAMPLE 

A  part  of  $  100  was  lent  at  5  %  annually,  and  the  rest  at 
7%  annually;  the  total  interest  for  one  year  was  $5.50. 
What  was  each  sum  ? 

Solution.      1.   .05  x  +  .07(100  -  x)  =  5.50. 

2.  6x  +  7(100  -x)  =  550.     Multiplying  (1)  by  100. 

3.  .'.2x  =  150,  and  x  =  15. 

4.  The  parts  were  $75  and -$25. 

WRITTEN   EXERCISES 

1.  .5a;  +  8  =  20.5. 

2.  .05  a; -6.25  =  -3.75. 

3.  .15  ic  +  . 03  a.' =  .25  a; -2.1. 

4.  3.5a;-1.6a;  =  3a;-.75. 

6.   1.01  a; +  2.005 -.003  a;  =  45.306. 
6.   4.2  (3  a;  -  1.06)  -  .04  (3.5  x-5)  =  20.668. 
.1  a;  +  .05      .01  X  +  .005  ^  ^^^^^ 
2  3  '         ^' 


EQUATIONS  173 

.03(5-.4a;)      .7(2a^-i)^gg^^ 
.2  ^         .5  '       " 

9.  A  bank  lent  $500  in  two  parts,  one  at  6%  and  the 
other  at  4%.  The  annual  interest  on  the  S500  was  $25.90. 
What  was  each  part  ? 

10.  In  a  certain  solution  of  camphor  and  alcohol,  the 
amount  of  camphor  was  25  %  of  the  amount  of  alcohol.  How 
much  of  each  in  4.5  oz.  of  the  mixture  ? 

11.  In  a  certain  grade  of  concrete  composed  of  pure  cement 
and  gravel  the  weight  of  the  cement  is  66|  %  of  the  weight  of 
the  gravel.  How  many  tons  of  pure  cement  are  there  in  a 
block  of  this  concrete  weighing  6.6  T.  ? 

237.  The  Linear  Form  ax  +  b.  Every  polynomial  of  the 
first  degree  can  be  put  into  the  form  ax  4-  h.  That  is,  by  re- 
arranging the  terms  suitably,  it  can  be  written  as  the  product  of  x 
by  a  number  not  involving  x,  plus  an  absolute  term.  Hence,  the 
form  ax-\-b  i^  called  a  general  form  for  all  polynomials  of  the 
first  degree  in  x. 

Thus,  4  a;  +  3(2  —  5  a;)  can  be  written  —  11  a:  +  6  which  is  in  the  form 
of  ax  +  6,  because  —  11  takes  the  place  of  a  and  +  6  takes  the  place  of  h. 

238.  General  Equations.  Just  as  every  linear  polynomial  is 
of  the  general  form  ax  -\-  b,  so  every  equation  of  the  first 
degree  is  equivalent  to  an  equation  of  the  form 

ax  -{-  6  =  0; 

consequently  the  latter  is  called  a  general  equation  of  the  first 
degree  ivith  one  unknown. 

239.  General  Solution.     From  the  equation  ax-\-b  =  0,  (1) 

we  have  ax=  —  b,  {2) 

and  hence,  x  =  ^^'  (3) 

a 

240.   is  the  general  form  of  the  root  of  the  equation  of 

the  first  degree.     There  is  always  one  root,  and  only  one. 


174  A  HIGH   SCHOOL   ALGEBRA 

241.    To  solve  an  equation  of  the  first  degree  in  an  unknown : 

Put  the  equation  into  the  form  ax  +  b  =  0.     Tlien  divide  the 
absolute  term,  h,  with  its  sign  changed  by  the  coefficient  of  x. 

WRITTEN   EXERCISES 

Simplify  and  solve  according  to  Sec.  241 : 

1.  l  =  4a;(3  +  7).  ^     a;  +  1^4^ 

2.  6+8(1-.t)  =  2.  "        ^         ^' 

3.  5x  +  3(a;-2)  =  4a;.  ^        ^    =2. 


4.   a  =  bx  +  c.  -'-"'"^ 


5. 


1_3  9.    a  =  i^+i. 


^_j_^  10.   Spx+ Abd=7ax—b+d 

^'    'x  +  2^^'  11.    to(l-a)+a(w;-l)  =  0. 

12.-   5.  +  15  +  ^^±i5  =  ^^±^. 

13.  3(aj  +  l)  +  7(a;  +  2)=4(a;  +  3). 

14.  {x  —  a){x  —  b)  =  x^  —  a. 

15.  ^bx  —  l(x -\-b)  +  ac—cx  =  0. 

16     i      ^  —  x^^a  +  x 
2  4 

17.    2[3(a'+l)+l]+l  =  0. 
18.    -^ ^L_  =  o.  19.    (3-x)(7-a;)  =  10+ar^. 

20.  a;(l  +  a)  +  2  a;(l  +  5)  =  3  .^(1  +  c). 

21.  (l  +  3a:)2  +  (3  +  4a;)2  =  (l-5aj)2. 

22.  (l-a;)(4  +  a;)4-(2  +  a^)(6  +  a;)  =  0. 

23.  {a-x)(b-\-x)  =  c^-a^. 

24.  (a;  +  6)(x  +  d)=(aj-hl)l 

25.  _l-4.±i^^=^_H-^izi^. 
a  -\-b         X         a  —  b         x 


EQUATIONS  175 

Solve  for  a : 

26.    a&+ac  =  4.  27.    ag -\-3  =  2ah- 5, 

28.    (a-l)(a  +  2)  =  {a-h){a-\-h). 

29.  Solve  this  equation  for  e ;  also  for  h  : 

e  e 

Solve  for  ^ : 

30.  v  =  at  +  ^-^'  31.   gh  +  gh^  =  r^. 

32.  How  much  alcohol  must  be  added  to  2  qt.  of  a  solution 
90  %  pure  alcohol  to  make  a  solution  95  %  pure  ? 

Suggestion.     Let  x  equal  the  number  of  quarts  of  alcohol  to  be  added. 

Then,  the  new  solution  will  contain  (2  +  x)  qt.  The  water  in  the  new 
solution,  or  5%  of  (2  +  x)  qt.,  will  be  the  same  as  that  in  the  original 
solution,  or  10%  of  2  qt. 

33.  18  %  by  weight  of  wheat  is  lost  (as  bran,  etc.)  in  grind- 
ing it  into  flour.  How  many  60-lb.  bushels  of  wheat  are  used 
in  making  738  lb.  of  flour? 

34.  Divide  the  number  n  into  two  parts  A^  B,  such  that  A 
shall  be  f  of  5. 

35.  A  train  leaves  A  for  B  100  mi.  distant,  traveling  40  mi. 
per  hour.  At  the  same  time  a  train  leaves  B  for  A,  traveling 
45  mi.  per  hour.  How  far  from  A  will  they  meet,  and  how 
long  after  starting? 

I  40  a;  ,  45  a;  I 


A  100  mi.  B 

Solution.     1.    Let  x  be  the  number  of  hours  from  the  time  the  trains 
start  until  they  meet. 

2.  Then,  the  train  from  A  travels  40  x  mi.,  and  the  train  from  B,  45  x  mi. 

3.  Therefore,      40ic  +  45x  =  100,  as  in  the  figure. 

4.  Therefore,  x  =  V/  =  ItV  the  number  of  hours. 

5.  And  40  X  =  47 j^^^,  the  number  of  miles  from  A. 
Check.     Find  the  distance  from  B  and  add  to  47jV  Dii» 


176         A  HIGH  SCHOOL  ALGEBRA 

36.  A  train  leaves  A  for  a  station  B,  d  miles  distant;  its 
rate  is  r  miles  per  hour ;  h  hours  later  another  train  leaves  B 
for  A,  running  E  miles  per  hour.  How  far  will  each  train 
have  traveled  when  they  meet  ? 

Solution.  1.  Let  x  =  the  number  of  hours  traveled  by  the  first  train 
before  they  meet. 

2.  Then,  x  —  h  =  the  number  of  hours  traveled  by  the  second  train. 

3.  rx  =  number  of  miles  traveled  by  first  train. 

4.  B(x  —  h)=  number  of  miles  traveled  by  second  train. 

5.  .*.  rx  -\-  B(x  —  h)  =  d,  by  the  conditions  of  the  problem. 

6.  .*.  (r  +  Il)x  =  d  +  Bh,  performing  the  operations. 
n      .    ^      d-\-Bh 


r  +  B 


,  solving  (6). 


8.  .\  rx=:  ^ — — ^ ,  the  distance  traveled  by  the  first  train. 

r  +  B 

9.  And  B(x-  h)  =  -^^^~'^^\  the  distance  traveled  by  the  second 

r  +  B 
train. 

37.  Read  the  above  problem,  taking  d  =  450,  r  =  50,  and 
B  =  40,  and  h  =  41  Solve  the  problem  by  substituting  these 
numbers  in  the  expressions  of  steps  8  and  9. 

242.  Formula.  An  expression  which  shows  how  a  desired 
number  is  to  be  obtained  from  given  numbers  is  called  a 
formula. 

Thus,  the  expressions  in  steps  8  and  9  of  Problem  36  are  formulas. 
A  formula  contains  the  solution  of  all  problems  which  may  be  made  by 
simply  changing  the  numbers  in  the  given  problem. 

WRITTEN    EXERCISES 

1.  The  sum  of  three  numbers  is  40.  The  second  is  6  more 
than  the  first,  and  the  third  is  the  sum  of  the  other  two. 
Find  the  numbers. 

2.  The  sum  of  three  numbers  is  s ;  the  second  is  a  greater 
than  the  first,  and  the  third  is  the  sum  of  the  other  two. 
Find  each  number. 

3.  The  sum  of  three  numbers  is  s.  The  second  is  a  less 
than  the  first,  and  the  last  is  b  less  than  the  second.  Find  the 
numbers. 


EQUATIONS  177 

4.  Apply  Exercise  3  to  the  problem  :  The  sum  of  three  num- 
bers is  35.  The  second  is  3  less  than  the  first,  and  the  third 
is  4  less  than  the  second.     Find  the  numbers. 

243.  Special  Problems.  The  solution  of  many  problems  is 
made  easier  by  a  proper  selection  of  the  unknown  quantities. 

WRITTEN    EXERCISES 

1.  A  certain  number  consists  of  two  digits  whose  difference 
is  3 ;  and,  if  the  digits  are  interchanged,  the  number  so  formed 
is  ^  of  the  original  number.     Find  the  latter. 

Suggestion.     1.   Let  x  be  the  smaller  digit. 

2.  Then  3  +  a:  is  the  greater. 

3.  .'.  10(cc  +  3)  +  X  is  the  given  number. 

4.  .'.  10  X  +  (x  +  3)  is  the  number  with  the  digits  interchanged. 

5.  According  to  the  problem,  10  x  +  (x  +  3)  =  f  [10  (x  +  3)  +  x]. 

6.  Solve  for  x. 

2.  The  tens'  digit  of  a  certain  number  exceeds  the  units' 
digit  by  4,  and  when  the  number  is  divided  by  the  sum  of  its 
digits,  the  quotient  is  7.     Find  the  number. 

3.  A  boatman  who  rows  3|-  mi.  in  still  water  finds  that  it 
takes  him  12  hr.  to  row  upstream  a  distance  that  he  can  row 
down  in  2  hr.     What  as  the  rate  of  the  current  ? 

Suggestion.  Let  x  be  the  rate  of  the  current.  Then,  the  boatman 
rows  upstream  at  the  rate  of  3|  —  x  miles  per  hour,  and  downstream  at 
the  rate  of  3^  -f  x  miles  per  hour. 

4.  The  sum  of  three  numbers,  a,  6,  c,  is  3036;  a  is  the 
same  multiple  of  7  that  h  is  of  4,  and  also  the  same  multiple 
of  5  that  c  is  of  2.     What  are  the  numbers? 

7x 

Suggestion.     Show  that  a,  6,  and  c,  may  be  denoted  by  7x,  4  x, 2. 

5 

5.  A  number  is  composed  of  three  digits  each  greater  by 
one  than  that  on  its  right ;  the  difference  between  the  number 
and  \  of  the  number  formed  by  reversing  the  order  of  the 
digits  is  36  times  the  sum  of  the  digits.     Find  the  number. 

Suggestion.     A  convenient  selection  of  the  digits  is  x  +  1,  aj,  x  —  1. 


178  A  HIGH  SCHOOL  ALGEBRA 


REVIEW 
WRITTEN    EXERCISES 


Solve  for  x 


1    a;  +  l  ,  '^  +  3^2 

x-^2     ic  +  4 

2.  {a-x){b-\-x)  =  h-y?. 

3.  (o  +  i«)(6  4-a^)  =  «&  +  a^. 

4.  (rt_a;)(6-a!)  =  6-ha^. 

5.  (a  +  5a^)(a-6ic)  =  (a2  +  62a;)(a_a;). 

6.  (a  +  6x-)  (5  -  ax)  =  {a  —  hx)  {ax  —  b). 

7.  (l-x){2-x)-{S-x){4.-x)  =  5-x. 

8.  (a -{- x)(b  —  x)  =  a  —  x^. 

9.  aa;(&a;  +  1)  =  6a;(aa;  + 1)  -f  a  +  6. 

10.  (1  +  a;)(2  -f  aj)  4-  (5  +  6a.')(l  +  8a;)  =  (4  +  7xy 

11.  (a  +  a;)(c?  +  x)  — (c  — i»)(a  — a;)  =  0. 

12.  How  much  water  must  be  added  to  80  lb.  of  a  5  % 
solution  of  salt  to  obtain  a  4  %  solution  ? 

13.  Two  men  are  25  mi.  apart  and  walk  toward  each  other  at 
the  rates  of  31  mi.  and  4  mi.  an  hour  respectively.  After  how 
long  do  they  meet  ? 

14.  A  man  travels  50  mi.  in  an  automobile  in  S\  hr.  If  his 
car  runs  at  the  rate  of  20  mi.  an  hour  in  the  country,  and  at  the 
rate  of  8  mi.  an  hour  when  within  city  limits,  find  how  many 
miles  of  his  journey  are  made  in  the  country. 

15.  A  train  running  30  mi.  an  hour  requires  21  min.  longer 
to  go  a  certain  distance  than  does  a  train  running  36  mi.  per 
hour.     What  is  this  distance  ? 

16.  A  and  B  set  out  on  an  automobile  trip,  A  having  |  as 
much  money  with  him  as  B;  after  A  had  paid  out  $1  less 
than  -|  of  his  money,  and  B  had  paid  $  1  more  than  ^  of  his,  it 
was  found  that  B  had  left  only  half  as  much  as  A.  How 
much  had  each  at  the  outset  ? 


EQUATIONS  179 

17.  In  forming  a  regiment  into  a  solid  square  60  men  were 
left  over ;  but,  when  formed  into  a  rectangle  with  5  men  more  in 
front  than  before  and  3  less  in  depth,  there  was  one  man  want- 
ing to  complete  it.  Find  the  total  number  of  soldiers  in  the 
regiment. 

18.  Two  automobiles  start  from  the  same  place ;  one  goes 
east  at  the  rate  of  18  mi.  an  hour  and  the  other  west  at  15  mi. 
an  hour.     In  how  many  hours  are  they  330  mi.  apart  ? 

19.  A  train  going  from  New  York  to  Chicago  at  the  average 
rate  of  40  mi.  an  hour  takes  44  hr.  longer  than  one  going  50  mi. 
an  hour.     Find  the  distance  between  these  places. 

20.  Given  a  =  the  amount,  r  =  the  rate,  and  /  =  the  prin- 
cipal in  a  problem  of  simple  interest,  what  is  the  time  ? 

21.  In  Exercise  20  let  a  =  $560,  r  =  4%,  and  ^  =  3  yr. 
Eind  the  principal. 

22.  Divide  the  number  a  into  two  j)arts  such  that  m  times 
the  greater  may  exceed  7i  times  the  less  by  h. 

23.  A  certain  number  consists  of  two  digits  whose  difference 
is  5 ;  and,  if  the  digits  are  interchanged,  the  number  so  formed 
is  f  of  the  original  number.     Find  the  latter  number. 

24.  A  certain  numbeiv  consists  of  two  digits  of  which  the 
tens'  digit  is  3  times  the  units'  digit;  and,  if  these  digits  are 
interchanged  a  number  is  formed  which  is  less  than  the  original 
number  by  36.     Find  the  original  number. 

25.  Separate  275  into  two  parts  such  that  \  of  the  smaller 
equals  \  of  the  larger. 

26.  The  sum  of  three  consecutiver  even  numbers  is  306. 
Find  the  numbers. 

27.  One-fourth  of  a  man's  age  now  equals  \  of  what  it  was 
20  yr.  ago.     Find  his  age  now. 

28.  In  a  certain  number  of  two  digits  the  tens'  digit  exceeds 
the  units'  digit  by  2 ;  if  the  number  is  diminished  by  f  of  the 
sum  of  its  digits,  they  are  interchanged.     Find  the  number. 

29.  Separate  150  into  two  numbers  such  that  if  one  be  divided 
by  23  and  the  other  by  27,  the  sum  of  the  quotients  is  6. 


180  A   HIGH   SCHOOL   ALGEBRA 

30.  The  current  of  a  certain  river  is  6  mi.  per  hour.  A 
steamer  can  go  upstream  in  9  hr.,  a  distance  that  takes  only 
3  hr.  to  come  down  using  the  same  power  per  hour.  Eind 
the  rate  of  the  steamer  in  still  water. 

31.  A  boatman  who  could  row  5  mi.  per  hour  in  still  water 
rowed  a  certain  distance  up  a  stream  and  back  again.  The 
current  was  3  mi.  per  hour,  and  it  took  the  boatman  10  hr.  to 
make  the  round  trip.     How  far  up  the  stream  did  he  go  ? 

32.  A  can  do  a  piece  of  work  in  3  da.  But  A  and  B  working 
together  can  do  the  same  work  in  2  da.  How  many  days  would 
it  take  B  to  do  it  alone  ? 

Suggestion.  Let  x  —  the  number  of  days  required  by  B.  Then,  A 
can  do  -  in  1  da.  and  B  can  do  -  in  I  da.  The  sum  of  these  multiplied 
by  2  will  be  the  whole,  or  1. 

33.  To  a  mass  of  metal  composed  of  4  parts  of  silver  to  1 
part  of  tin,  enough  silver  is  added  to  make  a  mass  containing 
6  parts  of  silver  to  1  part  of  tin.  How  many  ounces  of  silver 
are  added  per  ounce  of  the  original  metal  ? 

34.  Twenty  coins  composed  of  dimes  and  quarters  amount 
to  %  3.20.     How  many  coins  of  each  kind  are  there  ? 

SUMMARY 
The    following    questions    summarize   the   definitions   and 
processes  treated  in  this  chapter: 

1.  How  is  the  degree  of  an  equation  determined  ?     Sec.  225. 

2.  Define  and  illustrate  linear  equation ;  also  quadratic 
equation  ;  also  higher  equation.  Sees.  226-228. 

3.  What  is  the  absolute  term  ?  Sec.  230. 

4.  What  is  the  first  step  in  the  solution  of  a  fractional 
equation  ?  Sees.  231-233. 

5.  How  are  decimal  coefficients  best  removed  ?         Sec.  236. 

6.  What  is  the  general  form  of  an  equation  of  the  first  de- 
gree ?     What  is  the  form  of  the  root  ?  Sees.  238-240. 

7.  What  is  a /ormwZa.?  Sec.  242. 


CHAPTER   XVII 

RATIO,   PROPORTION,    AND   VARIATION 
RATIO 

244.  Ratio.  The  quotient  of  two  numbers  of  the  same  kind 
is  often  called  their  ratio. 

The  following  examples  illustrate  the  use  of  the  word  "ratio  "  : 

A  solution  consists  of  sulphuric  acid  and  water  in  the  ratio  of  2  to  3. 
This  means  that  |  of  the  whole  is  sulphuric  acid  and  |  is  water. 

Sterling  silver  requires  a  little  other  metal  (alloy)  to  harden  the  silver 
in  it ;  the  ratio  by  weight  of  the  amount  of  pure  silver  to  the  entire  mass 
is  usually  925  to  1000.     This  means  that  ^^^s^^  of  the  whole  is  silver. 

The  specific  gravity  of  a  solid  is  the  ratio  of  its  weight  to  the  weight  of 
an  equal  volume  of  water  under  standard  conditions. 

The  birth  rate  per  annum  in  a  city  is  said  to  be  23  when  the  ratio 
of  the  total  number  of  births  in  a  certain  year  to  the  total  number  of  in- 
habitants at  the  begin-  ^  ^___ . 

ning   of   that  year  is   — --s======:ririIIZIZZZIZZZZZIZZZZ__. i 

that  of  23  to  1000.  loo  ft. 

A  road  bed  is  said  to  have  an  8  %  grade  when  the  ratio  of  the  vertical 
rise  to  the  horizontal  distance  is  that  of  8  to  100. 

245.  Ratio  is  commonly  expressed  by  a  fraction. 

Thus,  the  ratio  of  2  to  3  is  written  f  ;  the  old  form  2 : 3  is  less  con- 
venient in  calculation. 

Before  ratios  were  written  in  fractional  form  it  was  convenient  to  have 
names  for  the  terms.  Thus,  the  first  number  was  called  the  antecedent^ 
and  the  second  number  the  consequent. 

In  division  the  divisor  may  be  abstract  or  concrete.  If  it  is  abstract, 
the  quotient  is  of  the  same  character  as  the  dividend.  If  it  is  concrete, 
the  dividend  must  be  concrete  and  expressed  in  the  same  unit. 

Consequently,  we  cannot  speak  directly  of  the  ratio  of  12  gal.  to  3  qt. ; 
we  must  first  express  both  numbers  in  the  same  unit,  as  48  qt.  to  3  qt. 

Similarly,  when  we  speak  of  the  ratio  of  the  distance  to  the  time,  we 
mean  the  ratio  of  the  corresponding  abstract  numbers,  as  in  Sec.  244. 

181 


182  A  HIGH  SCHOOL  ALGEBRA 

WRITTEN    EXERCISES 

1.  How  much  pure  silver  is  there  in  200  oz.  of  sterling 
silver  (Sec.  244)  ? 

Suggestion.     —  =  -^  • 
200      1000 

2.  A  silversmith  buys  46J  oz.  of  pure  silver;  how  much 
sterling  silver  can  be  made  from  it  ? 

Suggestion.     The  equation  is  — ^  = 

X       1000 

3.  What  is  the  rate  per  second  of  a  train  which  travels 
uniformly  635  ft.  in  5  sec.  ? 

4.  The  weight  of  a  piece  of  gold  is  94.5  oz.,  and  the  weight 
of  an  equal  volume  of  water  is  5  oz.  What  is  the  specific 
gravity  (Sec.  244)  of  the  gold  ? 

5.  There  are  4053  births  in  a  certain  city,  making  its  birth 
rate  21  (Sec.  244).     What  is  the  population  of  the  city  ? 

6.  If  the  population  of  a  city  is  p  and  the  birth  rate  is  b, 
indicate  the  number  of  births. 

7.  The  top  of  a  mountain  pass  is  1200  ft.  vertically  above 
the  level  of  the  base ;  the  top  is  reached  by  a  zigzag  road  5  mi. 
long.     What  is  the  average  grade  of  the  road  ? 

8.  A  road  m  mi.  long,  measured  horizontally,  ascends  to 
a  height  of  /  ft.  above  the  level  of  its  starting  point.  Indi- 
cate the  average  grade  of  the  road. 

9.  Two  men,  A  and  B,  divide  $  963  of  profits  so  that  A's 
part  is  to  B's  in  the  ratio  of  2  to  1.  How  many  dollars  has 
each? 

Suggestion.     1.    Let  x  be  the  amount  A  receives. 

2.   Then,  963  —  a;  is  the  amount  B  receives. 

r  2 

^     '  =-,  the  ratio  of  the  shares,  as  given. 


963  -  x     1 
4.    Clear  the  equation  of  fractions  and  solve  for  x. 

10.    Tavo  partners,  A  and  B,  divide  $575  in  the   ratio   of 
2  to  3.     How  many  dollars  does  each  receive? 


RATIO,   PROPORTION,   AND   VARIATION  183 

246.  Special  Notations.  When  a  letter  stands  for  different 
values  of  the  same  variable  quantity,  these  values  are  commonly 
designated  either  by : 

1.  Small  letters  and  capital  letters. 

For  example,  if  we  have  to  represent  the  rates  of  two  moving  bodies  in 
the  same  problem,  we  may  use  r  for  the  rate  of  one  and  B  for  the  rate  of 
the  other, 

2.  Primes  and  seconds. 

For  example,  if  two  weights  occur  in  the  same  problem,  we  may  ex- 
press one  by  lo'  and  the  other  by  w" ,  read  :  "  lo  prime  "  and  "  w  second." 

3.  Subscripts. 

If  a  problem  has  three  quantities  indicating  time,  we  may  express  these 
by  ?i,  t2i  ts,  read:  "f  sub-one,"  "^  sub-two,"  " f  sub-three,"  or  simply 
"«-one,"  "«-two,"  and  "^-three." 

Primes  and  subscripts  should  be  carefully  distinguished  from  exponents 
or  coefficients,  because  they  have  no  significance  other  than  to  distinguish 
numbers  of  the  same  kind  one  from  another. 


WRITTEN    EXERCISES 

1.  Solve  for  r  the  equation  -  =  — • 

3    a  * 

2.  Solve  the  equation  in  Exercise  1  for  B. 

1      r" 

3.  Solve  for  r'  the  equation  —  =  — 

r      5 

4.  Solve  the  equation  in  Exercise  3  for  r". 

5.  Solve    -^  =  ~    for  t. ;  also  for  t^- 

.5       ^2 
5 

6.  Solve  3  ^1  =  -  tt2  for  ^2-     Solve  the  equation  for  t. 

6 

7.  It  is  known  that  in  triangles  whose  corresponding  angles 
are  equal,  the  corresponding  sides  have  the  same  ratio. 

In  the  figure,  what  is  the 
ratio  of  a  to  a'  ?    Of  20  to  a;  ?  ^o^  ,.,^ 

Find  X.     What  is  the  ratio  of 
y  to  180  ?     Find  y. 
13 


184  A   HIGH   SCHOOL   ALGEBRA 

Supply  the  numbers  to  fill  the  blanks  in  the  table 


8. 

9. 
10. 
11. 
12. 

Substance 

?o  =  weight 
of  substance 

w'  =  weight  of 

an  equal  volume 

of  water 

Specific  gravity  =  — 

Lead 

Oak 

Tin 

Coal 

Alcohol 

56.5  oz. 
12.75  oz. 

153.3    lb. 
18       T. 

13.6  oz. 

5  oz. 
15  oz. 
211b. 
10  T. 
17  oz. 

247.    A  property  of  ratio  : 

The  numerical  value  of  a  ratio  whose  numerator  is  less  than  its 
denominator  is  iiicreased  by  adding  the  same  positive  number  to 
both  terms. 

The  numerical  value  of  a  ratio  whose  numerator  is  greater  than 
its  denominator  is  decreased  by  adding  the  same  positive  number 
to  both  terms. 


1.  For,  let  -  be  the  given  ratio  and  x  the  given  positive  number, 
h 


2.    To  determine  whether  ^   or  — 
h         b 
second  ratio  from  the  first. 


is  the  greater,  subtract  the 


3.    Then, 


a  +  aj      ab  -\-  ax  —  ah  —  bx 


(a—  b)x 
b(b  +  x)' 


^      b-i-x  b{b+x) 

(1)  If  a  <  6,  the  last  ratio  is  a  negative  number,  therefore  the  ratio  of 

less  inequality  -  would  be  increased  by  adding  x  to  each  of  its  terms. 
b 

(2)  If  a>b,  the  last  ratio  is  a  positive  number,  therefore  the  ratio  of 

greater  inequality  -  would  be  diminished  by  adding  x  to  each  of  its  terms. 
b 

A  ratio  whose  numerator  is  less  than  its  denominator  was  formerly 
called  a  ratio  of  less  inequality. 

A  ratio  whose  numerator  is  greater  than  its  denominator  was  formerly 
called  a  ratio  of  greater  inequality. 


RATIO,   PROPORTION,   AND   VARIATION  185 


(3)  If  a  =  6,  the  last  ratio  is  zero,  therefore  the  ratio  - ,  if  equal  to 

unity,  would  be  unchanged  by  adding  x  to  each  of  its  terms. 

A  knowledge  of  this  property  will  remove  the  temptation  to  say  that 
adding  or  subtracting  the  same  number  from  both  terms  of  a  fraction  does 
not  alter  its  value. 


ORAL     EXERCISES 

1.  What  is  the  effect  on  |  of  adding  4  to  each  term  ? 

2.  What  is  the  effect  on  f  of  adding  5  to  each  term  ? 

3.  AVhat  is  the  effect  on  \  of  adding  a  to  each  term  ? 

4.  What  is  the  effect  on  J  of  adding  6  to  each  term  ? 

5.  What  is  the  effect  on  f  of  adding  c  to  each  term  ? 

6.  Which  is  the  greater,  f  or  f  ?     Also  |-  or  |  ?     Why  ? 

7.  Which  is  the  greater,  -  or  ^^^  ?     Why  ? 

h        6  +  1 

3.   Which  is  the  less,  -  or  ^^  ?     Why  ? 

9.   Which  is  the  less,  -  or  -^^  ?     Why  ? 

10.    Compare  — ^  and  ;  also,  ^~      and 


n  +  1  n  +  2  n  +  1  w  +  3 

PROPORTION 

248.  Proportion.     An  equation  between  two  ratios  is  called 
a  proportion. 

mi,263(z6&25      553 

Thus,  -  =  - ,     — ^  =  +—,     —  =  -,     -  =  -  are  proportions. 

A  proportion  is  usually  read  in  one  of  two  ways :  For  example,  -  =  - 

5       X 

is  read  "  3  is  to  5  as  a  is  to  x,"  or  "  the  ratio  three-fifths  equals  the  ratio 
a  over  x." 

249.  The  numbers  forming  one  of  the  ratios  are  said  to  be 
"  proportional  to  "  the  numbers  forming  the  other. 

Thus,  in  the  proportion  '^■^  —  \%  the  numbers  12  and  9  are  proportional 
to  20  and  15. 


186         A  HIGH  SCHOOL  ALGEBRA 

250.  The  terms  "  proportional ''  and  "  proportionally  "  are 
used  with  the  meaning  "  in  the  same  ratio." 

For  example : 

The  express  rate  from  Chicago  to  New  York  is  f  2.50  per  100  lb.,  the 
excess  above  100  lb.  being  charged  proportionally.  This  means  the 
charge  for  the  excess  has  the  same  ratio  to  the  excess  as  2.50  has  to  100. 

Of  two  men  in  business  one  furnished  f  of  the  capital,  and  the  other 
^  of  it.  They  divided  their  gain  of  $  9000  in  proportion  to  their  capitals. 
This  means  that  they  divided  the  $  9000  into  two  parts  having  the  ratio  of 
2  to  1. 

251.  The  expression  pro  rata  is  often  used  with  the  same 
meaning  as  "  proportionally  "  or  "  in  the  same  ratio." 

For  example  : 
A  and  B  hire  an  automobile  for  a  trip  and  agree  to  pay  |  and  |  of  the 
rental  respectively,  and  other  expenses  occurring  on  the  trip  pro  rata. 
This  means  that  the  other  expenses  are  to  be  divided  between  A  and  B  in 
the  same  ratio  as  the  rental  of  the  automobile. 

WRITTEN   EXERCISES 

1.  Two  families  of  3  members  and  5  members  respectively 
camp  out  together  at  an  expense  of  $  160  ;  they  divide  this 
amount  in  proportion  to  the  size  of  the  families.  How  much 
did  each  family  pay  ? 

2.  A  man  hires  a  piano  at  $  5  per  month  of  30  days,  and  is 
to  pay  pro  rata  for  any  part  of  a  month.  What  does  he  pay, 
if  he  keeps  the  piano  51  days  ?      d  days  ? 

3.  Three  farmers  share  in  the  purchase  of  a  steam  thresher. 
A  pays  $  400,  B  pays  $  600,  and  C  pays  $  1000.  In  the  course 
of  the  year  the  thresher  is  rented  to  other  farmers  27  days 
at  $  10  per  day,  and  the  earnings  divided  among  the  owners 
pro  rata.     What  does  each  receive? 

4.  If  the  thresher  of  Exercise  3  is  rented  d  days  at  r  dollars 
per  day  and  the  earnings  divided  pro  rata,  what  does  each 
owner  receive  ? 

5.  It  cost  It  dollars  to  repair  the  thresher,  and  the  cost  is 
divided  pro  rata  among  the  owners.     What  does  each  pay  ? 


RATIO,   PROPORTION,  AND  VARIATION  187 


WRITTEN    EXERCISES 

X      160 

1.  Find  X  in  the  proportion  —  =  -^'  (1) 

Solution  : 

^_  15x160  ... 

Multiplying  both  members  by  15,       ^  —  — •  K.'^j 

Simplifying  the  fraction  in  (2),  x  =  96.  (3) 

2.  Find  o;  in  the  proportion    -  =  '- — .  (1) 

ic       42  '  ^ 

Solution  : 

Clearing  of  fractions,  7  X  42  =  .  14  X.  (^) 

7  V  42 

Simplifying  (2),  X  =  ^-^^  =  2100.  (5) 


h      h' 

3.  Solve  the  proportion  -  =—  for  h.     For  h'.     For  V. 

c       I 

r)      W 

4.  Solve  the  proportion  ^  =  —  for  p.  For  P.  For  W,  For  w, 

5.  Solve  the  equation  ^  =  — ,  for  p.  For  h.   For  6'.  For  p'. 

6.  In  Exercise  5  what  is  the  value  of  p',  if  p  =  15,  h  =  28, 
and  V  =  30.5  ? 

7.  Two  partners,  A  and  B,  in  business  divide  $  9000  between 
them  in  the  ratio  of  2  to  1.     How  much  does  each  receive  ? 

8.  At  $  2.80  for  8  hours'  work,  overtime  paid  proportionally, 
how  much  does  a  workman  receive  for  2i  hr.  overtime  ? 

9.  After  rents  rose  \  and  other  things  in  proportion,  a 
family's  expenses  for  one  month  were  %  132.  What  were  their 
expenses  before  the  rise  ? 

10.  1  cu.  ft.  of  lime  and  2  cu.  ft.  of  sand  are  used  in  making 
2.4  cu.  ft.  of  mortar.  How  much  of  each  is  needed  to  make  72 
cu.  ft.  of  mortar  ? 

11.  In  making  glass,  1,  5,  and  15  portions  by  weight  of  chalk, 
potash,  and  sand  respectively  are  used.  How  many  pounds 
of  each  are  required  to  produce  1176  lb.  of  glass  ? 


188 


A  HIGH  SCHOOL  ALGEBRA 


252.   Problems  of  the  Lever : 

1.  The  point  of  support  of  a  lever  is  called  the  fulcrum  and  in  the 
figure  is  denoted  by  F. 

If  P  denotes  the  power  and  W  the  weight   (expressed  in  the  same 

unit),   and  if  p  denotes  the  length  of 
P     the  arm  FP  and  w  the  length  of  the 
p       -•'  \      arm  FW  (both  expressed  in  the  same 

A  unit),  then  it  is  known  that 


|wl 


w 


ML 


t  In  words  :   The  ratio  of  the  lengths  of 

i      the  arms  equals  the  reciprocal  of  the 

""  ratio  of  the  corresponding  forces. 

Express  w  in  terms  of  F,  W,  and  p.     Express  W  in  terms  of 
P,  p,  and  w. 

2.   Find  piiP  =  10  lb.,  W  =  60  lb.,  w  =  2  ft. 

Supply  the  numbers  to  fill  the  blanks  in  the  following  table 
concerning  levers  : 


P 

w  ■ 

P 

W 

3. 

(          ) 

20  in. 

1.60  lb. 

90  1b. 

4. 

1.5  ft. 

(         ) 

2.25  lb. 

1125  lb. 

5. 

12  ft. 

^ft. 

(        ) 

960  lb. 

6. 

.3  ft. 

12.9  ft. 

2.5  T. 

(        ) 

253.    Problems  of  Similar  Triangles : 

1.   Areas     of     similar     triangles 
(those  of  the  same  shape)  are  pro-         3^u 
portional    to    the    squares    of    the      y^  ^  \ 
lengths     of     their     corresponding 
sides.     Let  A  and  A'  be  the  areas  of  two  similar  triangles, 

and  a  and  a'  be  a  pair  of  corresponding  sides.     Then 


a' 


Solve  the  proportion  —  =  — 
A'      a 


for  A.     For  A'. 


A'      a'2 
For  a.     For  a'. 


RATIO,   PROPORTION,   AND   VARIATION  189 

2.  Express  the  ratio  of  a  to  a'  in  Exercise  1. 

3.  The  areas  of  two  similar  triangles  are  64  sq.  in  and  625 
sq.  in.     What  is  the  ratio  of  any  pair  of  corresponding  sides  ? 

4.  If  the  side  a  of  the  smaller  triangle  in  Exercise  3  is 
8  in.,  what  is  the  corresponding  side  a',  in  the  larger  triangle  ? 
If  the  side  h  of  the  smaller  triangle  is  5  in.,  what  is  V  ? 

5.  The  lengths  of  a  pair  of  corresponding  sides  in  two  sim- 
ilar triangles  are  3  ft.  and  8  ft. ;  the  area  of  the  larger  one  is 
640  sq.  ft.     What  is  the  area  of  the  smaller  one  ? 

VARIATION 

254.  If  related  numbers  change  so  as  always  to  remain  in 
the  same  ratio,  one  is  said  to  vary  as  the  other. 

For  example  : 

At  5  ^  per  pound,  the  amount  paid  varies  as  the  number  of  pounds 
purchased. 

If  a  body  moves  at  the  same  rate,  the  distance  varies  as  the  time  of 
motion. 

If  $100  is  placed  at  simple  interest,  the  amount  of  interest  varies 
as  the  time. 

255.  "Varies  as"  is  thus  seen  to  be  merely  another  expres- 
sion for  "  is  proportional  to  "  or  "  varies  proportionally  with." 

ORAL  EXERCISES 

1.  How  much  more  can  a  man  lift  with  a  lever  8  ft.  long 
than  with  a  lever  4  ft.  long?  What  is  the  weight  lifted 
proportional  to  ?     What  does  the  weight  vary  as  ? 

2.  How  much  more  does  $  10  earn  at  6  %  simple  interest  in 
one  year  than  $  5  ?  When  the  rate  and  time  are  the  same 
what  does  the  interest  vary  as  ? 

3.  When  the  principal  and  the  rate  are  the  same,  what  does 
the  simple  interest  vary  as  ? 

4.  An  automobile  is  running  at  a  uniform  rate.  The  dis- 
tance traveled  varies  as  what  number  ? 

5.  If  the  machine  in  Exercise  4  traveled  90  mi.  in  3  hr.,  how 
far  would  it  travel  in  5  hr.  ? 


190 


A  HIGH  SCHOOL  ALGEBRA 


WRITTEN   EXERCISES 

1.  When  the  grade  of  a  roadbed  is  8|-  %,  what  is  the  rise  in 
a  horizontal  distance  of  175  ft.  ? 

2.  What  weight  can  a  man  weighing  175  lb.  raise  with  a 
lever  5  ft.  long,  if  the  fulcrum  is  6  in.  from  the  end  acting  on 
the  weight  ? 

3.  What  weight  can  he  raise,  if  he  weighs  180  lb.,  the  lever 
being  4  a  ft.  long,  and  the  support  being  a  ft.   from  the  end 

acting  on  the  weight  ? 

4.  The  height  to  which  the  mercury  rises  in  a 
barometer  varies  as  the  pressure  of  the  air  on 
the  mercury.  Let  the  heights  at  two  readings 
be  h  and  h\  and  the  corresponding  pressure  p 

and  »' :  then  ^  =  — .     Express  p  in  terms  of  li,  h\ 
and  p .  ^ 

5.  In  Exercise  4,  let  two  readings  of  the  barom- 
eter be  h  =  28.5  and  h^  =  29.5,  and  the  corresponding  pressures 
be  p  and  p'  =  15  lb.     What  is  the  value  of  p  ? 


Supply  the  blanks  in  the  following 

table  of  readings : 

P 

P' 

h 

h' 

(6) 
(7) 
(8). 
(9) 

(       )lb. 
14.75  lb. 
15  lb. 
14.2  lb. 

14.5  lb. 

(   ) 

15.25  lb. 
14.8  lb. 

30  in. 
29.5  in. 

(       ) 
28.75  in. 

29  in. 

30  in. 
30.5  in. 

(       ) 

PROPERTIES  OF  PROPORTION 

256.  Means  and  Extremes.  The  middle  numbers  in  a  pro- 
portion are  called  the  means  and  the  end  numbers  the  extremes. 

Thus,  in  the  older  form  of  writing  a  proportion,  a  :b  r.cid,  b  and  c 
are  called  the  means  and  a  and  d  the  extremes.  When  so  written,  the 
proportion  is  read  "  a  is  to  &  as  c  is  to  cZ."     At  present,  it  is  more  cus- 


tomary to  use  the  form  - 


and  to  read  it  "  a  over  b  equals  c  over  d.' 


RATIO,   PROPORTION,   AND  VARIATION  191 


257.  Fourth  Proportional.  In  the  proportion  -  =  -,  the 
fourth  number,  d,  is  called  the  fourth  proportional. 

Numbers  that  are  said  to  form  a  proportion  must  be  placed  in  the  pro- 
portion in  the  order  in  which  they  are  given.  Thus,  if  3,  5,  10,  and  x  are 
said  to  form  a  proportion,  then  3  :  5  =  10  :  aj. 

258.  Third  Proportional.     In  the  proportion  -  =  -,  the  third 

b     c 

number,  c,  although  in  the  fourth   place,  is  called   the  third 
proportional  to  a  and  b. 

259.  Mean  Proportional.  In  the  proportion  -  =  -,bis  called 
the  mean  proportional  between  a  and  c.  ^ 

260.  Relation  to  the  Equation.  A  proportion  is  an  equation 
expressing  the  equality  of  two  ratios. 

EXAMPLES 

1.  Find  the  fourth  proportional  to  the  numbers  6,  8,  30. 

•  Q  QQ 

Let  X  be  the  fourth  proportional,  then,  by  Sec.  257,  -  =  — .  (i) 

8        X 

Multiplying  by  885,  6  a;  =  8  •  30.  (^) 

Dividing  by  6,  ^  iC  =  40.  (3) 

Therefore  40  is  the  fourth  proportional  to  6,  8,  30. 

2.  Find  the  third  proportional  to  5  and  17. 

Let  X  be  the  third  proportional,  then,  by  Sec.  258,  — 

Multiplying  both  members  by  11  x,  5  X 

Solving  (f),  X 

Therefore  57|  is  the  third  proportional  to  5  and  17. 


WRITTEN    EXERCISES 

1.  Write  m,  n,  and  p  so  that  p  shall  be  the  third  propor- 
tional to  m  and  n. 

2.  Write  m,  n,  and  p  so  that  n  shall  be  the  mean  propor- 
tional between  m  and  p, 

3.  Solve:   -^  =  ^.  4.    Solve:    t?!  = -^. 

X      4:  X         .09 


17 

rn 

X 

\-^j 

172. 

{2) 

H^-- 

=  57|. 

(3) 

192  A   HIGH   SCHOOL  ALGEBRA 

Solve: 


5. 

X          7 
11      1331* 

8. 

1      42 
6       x' 

6. 

7. 

X  -i- 16  =  ^  -T-  X. 
-24.-^x  =  2x^ 

-12. 

9. 

75        -X 
-X        3 

10.  Find  the  third  proportional  to  8  and  5. 

11.  Find  the  third  proportional  to  —  6  and  —  4. 

12.  Find  the  mean  proportionals  between  the  following 
numbers  :  9  and  16 ;   —  25  and  —  4. 

13.  If  a  sum  of  money  earns  $48  interest  in  5  yr.,  how 
much  will  it  earn  in  16  yr.  at  the  same  rate  per  cent  ? 

14.  A  city  whose  population  was  40,000  had  2500  school 
children;  the  total  population  increased  to  48,000,  and  the 
number  of  children  of  school  age  increased  proportionally. 
How  many  children  of  school  age  were  there  then  ? 

15.  What  number  must  be  added  to  each  of  the  four  num- 
bers, 5,  29,  10,  44,  to  make  the  results  proportional  ? 

261.  Relation  of  Means  to  Extremes.  In  any  proportion  the 
product  of  the  means  equals  the  product  of  the  extremes, 

For,    -  =  -,  and  multiplying  both  members  by  hd^  ad  —  he. 
b      d 

262.  Conversely,  If  the  product  of  two  numbers  equals  the 
product  of  two  other  numbers,  the  four  numbers  can  be  arranged 
in  a  proportion,  the  two  factors  of  one  product  being  the  means, 
and  the  two  factors  of  the  other  product  the  extremes. 

For,  a  ad  =  be,  divide  both  members  by  bd,  and  -  =  — 

b      d 
Also,  if  a^—b^  =  xy, 

a_±b  _     y 
X  a—b 

263.  If  ?  =  ^,  then^  =  ?^. 

b      d  a     c 

For,  if  two  numbers  are  equal,  their  reciprocals  are  equal.     Sec.  219. 
The  older  form  of  statement  still  sometimes  used  is  :  If  four  numbers 
form  a  proportion,  they  are  in  proportion  by  inversion. 


RATIO,   PROPORTION,   AND   VARIATION  193 

264.  If  ^  =  ^,  then?  =  ^. 

b     a  c     d 

For,  multiplying  both  members  by  -,    -•-=-.-, 

c      c  h      c  d 

and  canceling  b  and  c,  ^  =  ^. 

c      d 

The  older  form  of  statement  is :  If  four  numbers  form  a  proportion, 
they  are  in  proportion  by  alternation. 

265.  If  ?  =  ^,  then''  +  ^-^  +  ^ 


Also 


b     d'  b  d 

a  —  b     c  —  d 


For,  adding  +  1  to  both  members,  -+]=-+!. 

h  d 

Therefore,  performing  the  processes,  ^-^^ —  :=:     "*"    , 

h  d 

Similarly,  by  adding  —  1  to  both  members  of  the  given  proportion  the 
second  form  results. 

The  older  form  of  statement  is :  If  four  numbers  form  a  proportign, 
they  are  in  proportion  by  composition  or  by  division. 

266.  If  ^  =  ^,  then^  =  ^. 

b     d  a  —  b     c  —  d 

For,  from  Sec.  265,  ^-±-^  =  ^-±-^, 

'  b  d 

„„-,  a  —  b      c  —  d 

and  ——-=——. 

b  d 

Dividing  these  equations,  member  for  member, 

a  -{-b  _c  +  d 
a  —  b     c  —  d 

The  older  form  of  statement  for  this  is  :  If  four  numbers  form  a  pro- 
portion, they  are  in  proportion  by  composition  and  division. 

267.  The  mean  proportional  between  two  numbers  is  the 
square  root  of  their  product. 

For,  from  ^  =  ^ ,  Sec.  259,  we  find  b  =  Vac. 
b      c 


194         A  HIGH  SCHOOL  ALGEBRA 

268.  Series  of  Equal  Ratios.  The  following  theorem  con- 
cerning ratios  is  sometimes  used : 

In  a  series  of  equal  ratios  (a  continued  proportion) ,  the  sum  of 
the  numerators  divided  by  the  sum  of  the  denominators  is  equal 
to  any  one  of  the  ratios. 

Proof  :  Let  -  =  -  =  —  =  ••.  be  the  equal  ratios  and  r  be  their  common 
value.  b     d      f  ^^^ 

Then  ^  =  r,  -  =  r,  -  =  r,  and  so  on.  (2) 

b        '  cl       '  f        '  ^  ^ 

.'.  a  =  br,  c  =  dr,  e=fr,  and  so  on.  (3) 

.'.  adding  the  equations  in  (.?), 

(a  +  c  +  e  ...)  =  (&  +  (^  +/...)  r.  {4) 

.-.  solving  (^)  for  r,  ^  +  ^  +  "-"  =  r.  (5) 

,    a  -\-  c  -\-  e  '••        the  sum  of  the  numerators        «    ^^  c     „„j  ^^  ^„ 
.*.  — ■ = =  -  ,  or  -,  and  so  on, 

b  -{■  d  -\-  f  •'•      the  sum  of  the  denominators      b         d 
since  r  is  any  of  these  ratios.  (6) 

269.  These  properties  may  be  applied  to  certain  equations. 


EXAMPLES 

1.  Given  -  =  -;  show  that  = .  (1) 

b     d'  5b  5d  ^  ^ 

Multiplying  both  members  of  3a  _  3j^  x^v 

the  given  equation  by  3,  b    ~   d 

Dividing  (2)  by  -5,  -^  ..  -^-  •  (5) 

—  00      —  ba 
Applymg  Sec.  265  to  (3),  — = (^4) 

6b  od 

2.  Three  numbers  are  in  the  ratio  of  3  to  4  to  7  and  their 
sum  is  42.     Find  the  numbers. 

Solution  : 

1.  Let  3  a;  be  the  first  number,  then  4  x  and  7  x  are  the  others. 

2.  .-.  3  X  +  4  X  +  7  X  =  42,  by  the  conditions  of  the  problem. 
.  3.    .  •.  14  X  =  42  and  x  =  3. 

4.   The  numbers  are  9,  12,  and  21  by  steps  (1)  and  (5). 


RATIO,   PROPORTION, 

AND  VARIATION 

195 

Solve : 

2 
4 

x-2       2 
-9x      9x 

(i) 

Applying  Sec.  268, 

2x_   2 
4       9x' 

(e) 

Simplifying  (?), 

x_   2 
2      9x* 

(3) 

Solving  (3), 

9x2  =  4, 

(4) 

and 

«:=±§. 

Test  by  substitution. 


WRITTEN    EXERCISES 

-     ^.         a      c      1        ,i,a— 46      c  —  4d 

1.  Given  -  =  -,  show  that  — ^ =  — 

b      d  4&  4.d 

2.  Given  ?  =  ^,  show  that      ^  ^ 


6d  b—ad—c 

3.    Given  ^  =  ^,  show  that    ^^         ^^ 


b      d  a—bc—d 

4.  Given  ^  =  ^  =  ^,  show  that  ^"^^~^=-. 

b      d      fl  b-\-d-f      d 

5.  Given  ^  =  ^  =  ^,  show  that  ^±4^  =  '4- 

6.  Given  !?^=^  =  r,  show  that  2m-3£+^-^m^ 

n      q      s  2  n-~3  q-\-o  s       n 

7.  Given  ?  =  *  =  i,  show  that  x  =  ^iHl^. 

a     c     d  c-\-d 

8.  Write  in  the  form  of  a  proportion  (Sec.  262)  : 

(1)  d'-\-2ab-\-b^  =  cd. 

(2)  d'-b^  =  a(b-\-c). 

(3)  a^-5a;4-6  =  7a;. 

ic  — 7      5fl7  +  7 


9.    rind  X  in 


a;  +  2      Sx 


10.   There  are  two  numbers  with  the  ratio  of  3  to  5  and  one 
is  24  less  than  the  other.     Find  the  numbers. 


196  A  HIGH   SCHOOL   ALGEBRA 

REVIEW 
ORAL  EXERCISES 

O  K 

1.  What  is  the  fourth  proportional  in  -  =  -  ? 

o      d 

q  K 

2.  What  is  the  third  proportional  in  -  =  -  ? 

5      X 

3.  What  is  the  mean  proportional  between  4  and  9  ? 

4.  State  the  principle  according  to  which  it  follows  from 

«  =  _^that^^±^  =  ^-±^. 
b      d  b  d 

5.  What  is  the  value  of  a;  in  -  =  -  ? 

a     X 


WRITTEN   EXERCISES 
Find  the  fourth  proportional  to : 
1.    X,  y,  ab.  2.  p^,  q,  m^q.  3.   x^,  ?/*,  z^. 

Find  the  mean  proportional  between : 
4.   a\  aW.  5.   16,  4  a^bK  6.   27  xy,  3  y. 


2a-Sb^2c-Sd 
36  Sd 

4a  —  56_4c  —  5cg 
5b       ~      5d 

a  —  b      b 


Nl 

len  -=-  show  that: 
b     d 

7. 

bd~  b""' 

8. 

a     a  —  b 
c     c  —  d 

9. 

a-\-2b     c-\-2d 

11. 


12. 


2b  2d  c-d     d 

13.  The  ratio  of  two  numbers  is  3  to  5  and  their  sum  is  48. 
Find  the  numbers. 

14.  The  lengths  of  the  sides  of  a  triangle  are  in  the  ratio  of 
3  to  4  to  5  and  their  sum  is  120  ft.  What  is  the  length  of 
each  side  ? 


RATIO,   PROPORTION,   AND   VARIATION  197 

15.   Denoting  by  b  the  arm  of  a  lever  on  which  the  power  p 
is  applied  and  by  a  the  arm  on  which  the  weight  iv  is  applied : 
What  force  is  required  to  raise  500  lb.,  if  a  =  2  and  b  =  10? 
Also,  if  a  =  8  and  6  =  40  ?     Also,  if  a  =  9  and  6  =  36  ? 

SUMMARY 

The  following  questions  summarize  the  definitions  and  pro- 
cesses treated  in  this  chapter  : 

1.  Define  ratio  ;  also  proportion.  Sees.  244,  248. 

2.  What  is  the  effect  of  adding  the  same  positive  number 
to  both  terms  of  a  fraction  ?  Sec.  247. 

3.  When  are  two  numbers  proportional  to  two  other  num- 
bers ?  Sec.  249. 

4.  Explain  '^ proportional,'^    ^' pro  rata"  and  "m  the  same 
ratio:'  Sees.  250,  251. 

5.  What  is  meant  by  "  one  quantity  varies  as  another  "  ? 

Sec.  254. 

6.  What  is  the  relation  of  variation  to  proportion  ? 

Sec.  255. 

7.  Define  the  extremes  and  means  of  a  proportion. 

Sec.  256. 

8.  What  is  the  fourth  proportional  to  three  numbers  ? 

Sec.  257. 

9.  What  is  the  third  proportional  to  two  numbers  ? 

Sec.  258. 

10.  What  is  the  mean  proportional  between  two  numbers  ? 

Sec.  259. 

11.  When  can  four  numbers  form  a  proportion  ?      Sec.  262. 

12.  Illustrate  the  result  of  taking  a  proportion  by  inver- 
sion ;  also  by  alternation  ;  by  composition ;  by  division ;  by 
composition  and  division.  Sees.  263-266. 

13.  State  a  property  of  a  series  of  equal  ratios.         Sec.  268. 


CHAPTER   XVIII 


GRAPHS   OF   LINEAR  EQUATIONS 

270.    Preparatory  : 

1.  If  $100  is  lent  at  2  %  simple  interest,  show  graphically 

how  the  amount  of 

-  interest  (/)  varies 
with  the  number  of 
years  (t). 

In  the  diagram  the 
spaces  on  the  horizontal 

•  scale  represent  the 
number  of  years,  and 
those   on    the   vertical 

■    scale    the    number    of 

.    dollars.     The  positions 

of    the    points  on   the 

line      OB     show     the 

•  amounts  of  interest  for 
.    the  periods  of  time  in- 
dicated on  the  horizon- 
tal scale.    Thus,  point 

«  ■•    P  shows  that   the  in- 
J   terest  is  $2  when  the 
time    is    1    yr. ;    point 
Q  shows  that  the  in- 
terest is  $4  when  the  time  is  2  yr. 

2.  What  does  point  B  show  ?     Point  0  ?     Point  S  ?     Point 
T? 

3.  What  does  the  point  halfway  between  Q  and  E  show  ? 

The  number  of  dollars  interest  is  always  twice  the  number 
of  years.     This  is  expressed  by  the  formula,  1=2  t. 

198 


3       4        5       6        7 
TIME  IN  YEARS 


GRAPHS   OF   LINEAR  EQUATIONS 


199 


From  the  graph  we  can  read  either  the  amount  of  interest 
for  a  given  time,  or  the  time  required  to  earn  a  given  interest. 

ORAL  EXERCISES 

From  the  graph  read  : 

1.  The  amount  of  interest  on  $100  at  2  %  for  2  yr.  For 
3  yr.     For  li  yr. 

2.  The  time  in  which  $  100  at  2  %  will  earn  $  4.     $  5.     $  3. 

WRITTEN    EXERCISES 

1.  Make  a  graph  like  that  on  page  198,  taking  the  rate  of 
interest  to  be  3%. 

2.  From  the  graph  answer  questions  like  1  and  2  above. 
State  the  relation  between  interest  and  time  in  this  case. 
Write  an  equation  expressing  this  relation. 

3.  Treat  similarly  each  of  the  rates :  4%;  2^%;  5%;  6%. 

4.  In  the  figure 
values  of  t  are  repre- 
sented on  the  hori- 
zontal line  OT,  and 
corresponding  values 
of  d  on  the  line  OD. 
What  is  the  value  of  t 
for  point  P?  Of  (^  for 
point  P?  What  is 
the  value  of  t  for  point 
Q?  Of  d  for  point  Q? 
Every  value  of  d  is 
what  part  of  the  corresponding  value  of  i?  Express  this 
relation  by  an  equation. 

,    5.    An  elevator  goes  up  at  the  rate  of  4  ft.  per  second. 


, 

1 

5 
^4 

) 

> 

2 

3^ 

Q2 

1 

Pg<^^^^^ 

^,^ 

T 

0        12        3        4       5 
TIME 

In  3 


sec. 


What 
In  5  sec.  ? 


distance  does  it  ascend  in  2  sec 
In  1  min.  ? 

6.    Letting  d  =  the  number  of  feet  passed  over  in  any  number 
of  seconds  (^),  and  using  horizontal  spaces  to  represent  the  num- 
ber of  seconds,  and  vertical  spaces  to  represent  the  number  of 
feet,  make  a  graph  to  represent  the  relation  c?  =  4 1 
14 


200 


A  HIGH  SCHOOL   ALGEBRA 


271.  Before  constructing  a  graph,  the  corresponding  values 
of  the  letters  may  be  conveniently  arranged  in  a  table  as  in 
the  following  example : 


Construct  the  graph  of 

2/  =  4  aj. 

Table 

X 

0 

y 

0 

1 

4 

2 

6 

8 

H 

10 

3 

12 

4 

16 

5 

20 

Graph 


8 

/ 

7 
6 
5 

/ 

••[ 1 

4 

/ 

\                \ 

3 

/ 

.J j 

2 

/ 

1 

/ 

0 

'         1:           2 

3       4        5       6 

WRITTEN   EXERCISES 

Construct  the  graphs  of : 


1.  2y=Sx. 

2.  4:y  =  x. 

3.  y  =  ix. 

4.  y  =  ix. 

5.  v  =  ^x. 


6.   iy  =  x. 


7. 

y  =  fx. 

8. 

y  =  lx. 

9. 

d  =  \t. 

10.    d  =  ^t. 


1. 

d  =  5t. 

2. 

\d=:t. 

3. 

d^S^t 

4. 

2^y  =  x. 

5. 

lix  =  y. 

272.    Graphs  may  be  constructed  for  negative  values  as  well 
as  for  positive  values  of  the  numbers  involved. 


EXAMPLES 

1.  If  we  regard  money  borrowed  (which  is  the  opposite  to  money  lent) 
as  negative,  and  interest  paid  (which  is  the  opposite  to  interest  received) 
as  negative,  we  may  express  by  a  single  line  the  changes  in  interest  and 
principal,  both  for  money  borrowed  and  for  money  lent. 


GRAPHS  OF  LINEAR  EQUATIONS 


201 


The  table  shows  the  change  in  interest  at  5%  for  2  yr.  corresponding  to 
the  change  in  the  principal  from  +  1 20  to  —  $  20,  the  positive  values  de- 
noting money  lent  and  interest  received,  the  negative  ones  denoting 
money  borrowed  and  interest  paid. 


Table 


p  = 

20 

10 

0 

-10 

-20 

i  = 

2 

1 

0 

-1 

2 

The  line  AOB  is  the  graph  representing  these  changes. 


r'^BbRROWEo""! 

j                                                                     5                     ! 

^  „. |....i 1. .  +  i ...i.-i .. J 

zzz^zzzMz. 

r lent! 1 

?:±E:^:i 

::::::^:::||^^ 

i4..i...j...U.t« 

i"i ill "■" 

+;»o_ 

2.   The  line  CB  in  the  figure  below  is  the  graph  of  the  equation  y  =  2Xy 


y 

X 

-4 

-2 

-2 

-1 

-1 

-i 

0 

0 

2 

1 

3 

^ 

4 

2 

for  both  positive  and  neg- 
ative values.  The  line 
i?0  is  the  graph  for  all 
positive  values  of  x  and  y 
and  the  line  OC,  the  ex- 
tension of  0J5,  is  the  graph  <  • 
for  negative  values.  The 
negative  values  of  x  are 
marked  off  to  the  left  of  0  and  the  negative  values  of  y  downward  from  0. 


202 


A    HIGH  SCHOOL   ALGEBRA 


273.  Any  change  that  takes  place  throughout  at  a  constant 
rate  is  said  to  take  place  "  uniformly." 

For  example,  a  body  moving  continually  at  the  same  speed  is  said  to 
move  uniformly.  That  the  graph  of  uniform  chaiim^vfiW  be  a  straight 
line  appears  from  the  fact  that  t\iQ  straight  line  is  the  only  path  along 
which  a  point  moves  upward  or  downward  uniformly.    Along  any  curved 


r 

^^^^ 

5 

^^.---^■''"^ 



\cQ 

/ 

,    A 

\ 

line  it  moves  upward  or  downward  more  rapidly  at  some  times  than  at 
others.  Thus,  in  the  figure  the  increase  from  J.  to  5  is  much  more  rapid 
than  from  B  to  C. 

If  X  and  y  vary,  subject  to  the  equation  ?/  =  4  x  +  3,  or  to  ?/  =  mx  +  &, 
the  change  will  be  uniform.  For,  in  the  first  case,  y  changes  by  4  times 
the  change  in  x,  and,  in  the  second  case,  by  m  times  the  change  in  x ; 
that  is,  y  varies  uniformly  with  x.  Since  we  know  that  only  straight 
lines  represent  uniform  change,  we  know  that  the  graphs  of  the  above 
equations  are  straight  lines. 


ORAL    EXERCISES 

1.  How  many  points  are  necessary  to  fix  the  position  of  a 
straight  line  ? 

2.  How  many  points  must  be  fixed  in  order  to  draw  a  graph 
which  is  known  to  be  a  straight  line  ? 

3.  When  several  points  are  constructed  corresponding  to  the 
values  of  the  unknowns  in  an  equation  and  a  straight  line  is 
drawn  through  any  two  of  them,  where  will  the  others  lie  ? 


GRAPHS  OF  LINEAR  EQUATIONS 


203 


274.    Constructing  the  graph  of  an  equation  is  called  plotting 
the  equation. 


WRITTEN    EXERCISES 

Plot  the  following  equations  : 


1. 

^y  =  X' 

2. 

y  =  3x. 

3. 

c  =  ip. 

4. 

y-i^' 

5. 

p  =  4.t. 

6. 

s  =  lt. 

7. 

2x-y  =  0. 

13. 

?/  =  ir  +  4. 

8. 

Qp  —  W  =  0. 

14. 

y=z5x. 

9. 

a;  -  6  ?/  =  0. 

15. 

y  =  -3x. 

10. 

y  =  2x^l. 

16. 

y  =  x. 

11. 

y  =  l-2x. 

17. 

y  =  ^x-i-6. 

12. 

y  =  x-4:. 

18. 

y  =  .Sx-2 

275.  Preparatory. 

1.  In  the  diagram  how  many  spaces  is  point  A  to  the  right 
of  line  yy'  ?     How  many  spaces  is  point  A  above  the  line  xx'? 

The  position  of  point  A  is  fixed  by  the 
distances  2,  1.  (It  is  customary  to  name 
the  distance  along  the  line  xx'  first. ) 

2.  How  many  spaces  is  point  B 
to  the  left  of  line  yy'  ?  How  many 
spaces  above  xx'  ? 

The  position  of  point  B  is  fixed  by  the 
distances  —  1,  2. 

3.  How  far  is  point  C  to  the  left 
of  yy'  ?     How  far  below  xx'  ? 

The  position  of  point  C  is  fixed  by  the  distances  —  1,  —  1. 

4.  How  far  is  point  D  from  yy'  ?     From  xx'  ?     What  dis- 
tances fix  the  position  of  this  point  ? 

The  position  of  point  D  is  fixed  by  the  distances  +2,  —  2. 

276.  Axes.     The  lines  of  reference  designated  by  xx'  and  yy' 
are  called  the  axes. 


■ 

y 

2 

1 

.A 

.-    1 

'"""] 

X     ■ 

^     -2      -ij      0 

1       \2 
1 

^.„ 

-1 

.J 

-2 

^D 

\ 

/' 

204 


A  HIGH   SCHOOL  ALGEBRA 


II 


277.  Quadrants.  The  axes  divide  the  diagram  into  four 
quarters  called  quadrants.  These  are  numbered  I,  II,  III,  and 
IV  as  shown  in  the  figure  in  Section  280. 

278.  The  position  of  the  point  P^  is  fixed  by  its  perpendicular 
distances,  P^M^,  PiN^,  from  the  axes.  These  two  perpendicular 
distances  are  called  the  coordinates  of  point  P^.  Similarly,  the 
coordinates  of  Pg  a-^e  P2J^2j  ^2-^2- 

The  position  of  any  point  is  fixed  by  its  two  coordinates. 

279.  The  point  of  intersection  of  the  axes  is  called  the 
origin  of  coordinates. 

280.  Abscissas  and  Ordinates.  The  distance  of  a  point  from 
the  axis  yy\  as  P^M^,  is  called  the  abscissa  of  the  point,  and 

the  distance  of  the 
y  -  point  from  the  axis 

N«  ^  xx'  is  called  the  ordi- 

Ni  Pi  nate  of  the  point. 

j ^  The  ordinates  of  all 

^i         ^i  points  in  quadrants    I 

and  II  are  positive,  and 
of  those  in  III  and  IV 
*  ^^  are  negative. 

^8  The  abscissas  of  all 

y^         '  '        '  points    in   quadrants   I 

and    IV    are    positive, 
and  those  of  all  points  in  II  and  III  are  negative. 

281.  Variables.  A  number  symbol  that  may  assume  differ- 
ent values  is  called  a  variable. 

Thus,  the  number  of  degrees  of  temperature  from  time  to  time,  the 
number  of  hours  from  sunrise  to  sunset,  the  price  of  w^h.eat,  the  number 
of  inhabitants  of  the  United  States,  are  variable  numbers.  They  meas- 
ure physical  or  other  quantities  that  are  in  themselves  variable, 

282.  Constants.  A  number  that  has  a  fixed  value  is  called 
a  constant. 

Thus,  3,  Vs,  —  4,  are  constants. 

283.  When  numbers  are  indicated  by  letters,  the  conditions 


III 


GRAPHS   OF   LINEAR   EQUATIONS  205 

of  the  problem  must  specify  which  are  constant  and  which  are 
variable. 

It  is  customary  to  use  the  earlier  letters  of  the  alphabet  to  denote 
constants,  and  the  later  ones  to  denote  variables,  but  it  is  not  necessary 
to  do  so. 

284.  Function.  When  the  value  of  one  variable  depends 
upon  that  of  ai;^ther  the  first  variable  is  said  to  be  a  function 
of  the  second. 

The  function  is  called  the  dependent  variable,  and  the  other 
the  independent  variable. 

For  example : 

1.  The  cost  of  a  railroad  ticket  depends  upon  the  number  of  miles 
to  be  traveled.  That  is,  the  cost  is  a  function  of  the  distance.  The 
distance  is  the  independent  variable,  and  the  cost  is  the  dependent 
variable.  Likewise,  the  distance  one  can  ride  depends  upon  the  cost  of 
the  ticket,  that  is,  the  distance  is  a  function  of  the  cost.  In  the  latter 
way  of  looking  at  it,  the  cost  is  the  independent  variable  and  the  distance 
the  dependent  variable. 

2.  If  a  train  moves  uniformly,  the  distance  traversed  is  a  function  of 
the  time. 

3.  An  iron  bar  expands  when  heated.  Its  length  is  a  function  of  the 
temperature. 

4.  3  X  —  2  is  a  function  of  x  because  its  value  depends  upon  that  of  x. 

285.  "  Function  of  «"  is  often  briefly  expressed  by /(a;). 


4, 


286.  Graphs  of  Functions.  The  graph  of  a  function  is  a  dia- 
gram representing  the  variation  of  the  function  due  to  the 
variation  in  the  value  of  its  variable. 

Thus,  the  first  graph  in  Sec.  271,  p.  200,  shows  that  the  value  of 
f(x)  =  4  X  varies  from  0  to  8  as  x  varies  from  0  to  2. 

287.  Since  the  graph  of  the  function  ax  -{-h  is  a  straight 
line  (Sec.  273),  this  function  is  called  a  linear  function,  and 
the  corresponding  equation,  y  =ax-\-h,  a  linear  equation. 


Thus,  if 

/(x)=:5x-4, 

then. 

/(2)  :=5.2-4  = 

/(O)  =5-0-4  = 

and 

/(r)  =  5r-4. 

206 


A  HIGH  SCHOOL  ALGEBRA 


288.  Graphs  of  Incomplete  Equations.  Either  xor  y  may  be 
lacking  in  an  equation  to  be  plotted. 

Thus,  the  equation  y  =  S  means  y  =  Ox-{-S,  in  which  y  =  S  for  all 
values  of  x  ;  consequently  the  graph  of  ^  =  3  is  a  line  parallel  to  the  x-axis 
and  3  units  above  it.  Similarly,  the  graph  of  x  =  2  is  a  line  parallel  to 
the  y-axis,  and  2  units  to  the  right  of  it. 


7.  0  =  «  +  3. 

8.  2x-2y  =  4:. 

9.  3x-{-Sy=9. 


i    1 


WRITTEN     EXERCISES 

Construct  the  graphs  of  : 

1.  ic  +  2/  =  1.  4:.   X  —  y  =  2. 

2.  x-i-y  =  0.  5.   x  =  4:y—l. 

3.  2y~4.x  =  2.  6.    Sx-\-2y=6. 

10.  What  line  must  the  graph  cross  when  x  =  0?  When 
y=0?  Construct  the  graph  of  ^  =  2  a;  —  3  by  using  the  points 
for  which  x  =  0,  and  y  =  0,  respectively. 

GRAPHS  OF  EQUATIONS  WITH  TWO  UNKNOWNS 

289.    Preparatory. 

1.  The  diagram  shows 
'  the  general  trend  in  the 
increase  of  the  pig-iron 
product  in  the  United 
States,  Germany,  and 
England  during  a  period  of 
ten  years. 

(a)  In  which  country 
has  the  increase  been  the 
most  rapid?  The  least 
-  rapid? 

(h)  In  what  year  was  the 
amount  produced  in  Eng- 
.  land  and  the  United  States 
the  same? 

(c)  Answer  the  same 
question  for  England  and 
Germany. 


GRAPHS   OF   LINEAR  EQUATIONS  207 

2.  A  factory  has  a  fixed  charge  of  $  20  daily,  and  makes  an 
average  gross  profit  of  $  2  daily  per  workman  employed.  What 
remains  after  the  fixed  charge  is  paid  is  net  profit.  Make  a 
graphic  representation  of  how  the  net  profit  (p)  varies  while 
the  number  of  workmen  (w)  varies  from  0  to  50. 

The  graph  exhibits  the  change  in  p  due  to  a  change  in  w  subject  to  the 
above  relation;  or  it  is  the  graph  of  the  equation  p  =  2  w  —  20. 

3.  A  second  factory  has  a  fixed  charge  of  $  60,  and  makes 
an  average  gross  profit  of  $3  daily  per  workman  employed. 
Construct  a  graph  to  represent  the  net  profits  of  the  second 
factory  in  the  diagram  made  for  Exercise  2. 

What  is  the  relation  between  p  and  w,  of  which  this  is  the 
graph  ? 

4.  From  the  diagram  read: 

(a)  The  number  of  workmen  for  which  each  factory  makes 
the  same  net  profit. 

(b)  The  amount  of  the  profit  mentioned  in  (a). 

(c)  The  number  of  workmen  beyond  which  the  second  factory 
makes  the  larger  net  profit. 

(d)  The  net  profit  of  each  factory  where  25  workmen  are 
employed.  40  workmen.  15  workmen,  10  workmen.  20 
workmen. 

(e)  The  number  of  workmen  each  factory  must  employ  to 
make  $  30  net  profit.     Also  $  60  net  profit.     No  net  profit. 

290.  Graphs  may  be  employed  to  represent  the  solution 
of  any  set  of  two  simultaneous  linear  equations  with  two  un- 
knowns. 

For  example : 

\x-V2y=A.  {2) 

We  have  an-eady  seen  how  to  represent  graphically  all  of  the  solutions 
of  a  single  equation  of  the  first  degree  in  two  unknowns. 


208 


A  HIGH  SCHOOL  ALGEBRA 


If  we  represent  the  equation  x-  y  z=l,  and  the  equation  a;  +  2  ?/  =  4, 
in  the  same  diagram  it  is  possible  at  once  to  read  the  valuer  of  x  and  y 
that  satisfy  both  equations. 


LE  OF  Values 
Equation  (1) 

X 

T 

If 
-1 

1 

0 

2 

1 

Table  of  Values 
FOB  Equation  (2) 


ORAL  EXERCISES 

1.  Eead  from  the  graph  of  the  first  equation  the  value  of  y 
when  x  =  0.     When  x  =  l.     When  x  =  2.     When  x  =  3. 

2.  How  many  points  are  needed  to  fix  the  position  of  a 
straight  line  ? 

3.  Kead  from  the  graph  of  the  second  equation  the  value  of 
y  when  x  =  0.     When  x  =  2.     When  ic  =  4. 

4.  What  point  in  the  diagram  is  common  to  the  two  graphs  ? 
What  are  the  values  of  x  and  y  for  point  s  ? 

5.  What  point  in  the  diagram  represents  the  solution  of  the 
system  of  equations  ? 

291.  The  solution  of  two  simultaneous  equations  of  the  first 
degree  in  two  unknowns  is  represented  graphically  by  the  point  of 
intersection  of  the  graphs  of  the  equations. 

Note.  Since  the  accuracy  of  the  graphical  solution  depends  upon  the 
precision  of  the  diagram,  the  results  so  found  must  be  regarded  as  approxi- 
mate. Their  accuracy  must  be  tested  in  the  usual  way.  The  graphical 
method  of  solution  is  of  practical  use  chiefly  in  applied  mathematics  where 
an  approximate  result  is  commonly  sufficient,  but  in  theoretic  mathematics 
it  is  important  as  exhibiting  clearly  to  the  eye  the  relations  between  the 
variables  involved. 


GRAPHS  OF   LINEAR   EQUATIONS  209 

WRITTEN    EXERCISES 


Solve  graphically : 


1. 

x-\-y  =  5, 

5. 

x^y  =  l, 

9. 

2x-y  =  l, 

Sx-2y  =  0. 

2x-y  =  5. 

«  -  2  2/  =  -  4. 

2. 

Sx-y  =  l, 

6. 

3x  +  y  =  12, 

10. 

4.x-{-y  =  6, 

x-\-2y^l2. 

x-y  =  0. 

2x-y=0. 

3. 

x-\-y  =  0, 

7. 

x-y  =  l, 

11. 

x-hSy  =  12, 

2/  +  3  =  4. 

2x-Sy  =  l. 

3x-y  =  (y. 

4. 

x  +  2y  =  Q, 

8. 

x  +  3y  =  S, 

12. 

x-iy  =  2, 

2a;-?/  =  2. 

x-2y  =  ~2. 

i^  +  2/  =  3. 

SUMMARY 

1.  In  graphical  work  what  are  the  axes  ?  The  quadrants  f 
The  coordinates  ?  The  origin  ^  The  abscissas  f  The  ordi- 
nates  ?  Sees.  276-280. 

2.  What  is  a  variable  f    A  constant  ?  Sees.  281,  282. 

3.  What  is  a  function  ?  What  is  meant  by  dependent  and 
independent  variables  f  Sec.  284. 

4.  What  does  the  graph  of  a  function  show  ?  Sec.  286. 

5.  When  two  linear  equations  are  plotted  on  the  same  axes, 
what  does  their  intersection  represent  ?  Sec.  291. 

HISTORICAL  NOTE 

We  have  stated  in  an  earlier  note  that  the  chief  symbols  of  algebra 
were  perfected  during  the  sixteenth  century,  and  that  scholars  had  then 
discovered  many  properties  of  algebraic  expressions  as  well  as  methods  of 
solving  equations.  But  at  the  beginning  of  the  seventeenth  century  a  new 
channel  for  algebraic  study  was  opened  by  the  French  philosopher  and 
mathematican,  Ren 6  Descartes,  who  laid  the  foundation  for  what  we  now 
call  "graphical  algebra." 

Ren6  Descartes,  was  born  at  La  Haye  in  1596.  At  the  age  of  twenty- 
one  he  enlisted  as  a  soldier  under  Prince  Maurice  of  Orange,  pursuing  the 
study  of  mathematics  in  his  leisure  hours.  In  1829  he  went  to  Holland, 
Dutch  culture  and  learning  then  being  at  its  height,  and  there  devoted 
twenty  years  to  the  preparation  of  his  works  in  mathematics  and  phi- 
losophy.    His  belief  in  the  certainty  and  accuracy  of  the  reasoning  used 


210 


A  HIGH  SCHOOL  ALGEBRA 


in  arithmetic  and  algebra  led 
him  to  apply  the  same  method 
to  physics  and  other  sciences. 
Descartes  adopted  the  plan  of 
locating  points  in  a  plane  by 
means  of  their  distances  from 
two  fixed  lines  or  axes,  and 
was  the  first  to  plot  lines  repre- 
senting algebraic  expressions,  as 
explained  in  Chapter  XVIII. 
The  words  "ordinate"  and 
"coordinates"  were  introduced 
by  him,  probably  from  the  Latin 
phrase  linece  ordinatce,  used  by 
the  Roman  surveyors  to  express 
parallel  lines.  The  word  ab- 
scissa was  first  used  by  an  Italian 
writer  (1659). 

Graphical  algebra,  in  a  broader 
sense,  was  used  many  centuries 

before  Descartes.    For  example,  the  products  : 

a(b  +  c)  =  ab  +  ac  and  (a  -{- by  =  a^ -\- 2  ab  +  b^, 

which  were  not  considered  direct  results  of  multiplication  before  Dio- 

phantos,  were  represented  graphically  by  Euclid  (300  b.c.)  in  the  following 

way: 


Kene  Descartes 


a(b  +  c) 


ah 

ac 

ah 

h' 

a- 

ah 

{a+by 


The  first  step  toward  constructing  graphs  of  equations  was  taken  by 
the  Hindoos  in  Bhaskara's  time,  when  they  represented  positive  andnega- 
tive  numbers  by  og^osite  segments  on  a^straightTine  ;  fromTthis  concep- 
tion it  was  onfyone  step  more'to  the  expression  of  the  relation  of  one 
variable  to  another  by  measurements  along  two  lines,  or  by  coordinates. 
This  was  done  by  Nicholas  Horem,  a  French  teacher  in  Paris  in  the  four- 
teenth century.  Naturally  he  began  with  positive  numbers  and  constructed 
graphs  in  the  first  quadrant  only,  so  it  remained  for  Descartes  to  show 
how  the  changes  in  a  function  of  x  can  be  shown  graphically  in  the  four 
quadrants  of  a  plane  for  all  real  values  of  x. 


CHAPTER   XIX 
SYSTEMS  OF   LINEAR  EQUATIONS 

EQUATIONS  WITH  TWO  UNKNOWNS 

292.  Problems  involving  more  than  one  unknown  often  re- 
quire more  than  one  equation  in  their  solution. 

EXAMPLES 

1.  A  street  car  of  a  certain  make  has  48  seats,  some  single 
and  some  double;  the  seating  capacity  of  the  car  is  56.  How 
many  single  seats  are  there  ?     How  many  double  seats  ? 

Solution.     Let  x  =  the  number  of  single  seats, 

and  y  =  the  number  of  double  ones. 

Then,                             I    x  +  y  =  48,  (1) 

\x  +  2y  =  66.  {2) 

Subtracting  equation  (J)  from      ,, o  /  o\ 

equation  (^),                    y  —  °'  w; 

Substituting  this  value  of„_i_Q_j.Q  /  /\ 

y  in  equation  (i),           X  +  b-^H.  {4) 

.:x=^0.  (5) 

Therefore  there  were  40  single  seats  and  8  double  ones.  (6) 

Test.  |      40+8  =  48. 

I  40 +  2. 8  =  56. 

10aj  +  52/  =  45,  '  (1) 

6x-\-Sy  =  4.7.  (^) 

Multiplying  (1)  by  3,  30  iC  +  15  y  =  135.  (3) 

Multiplying  (2)  by  5,  SO  Z  +  40  y  =  235.  (4) 

Subtracting  (3)  from  (A),  25  y  =  100,  and  y  =  4.  (5) 

Substituting  y  =  4  in  (1),  X  =  2|.  (6) 

Test  the  values  x  =  2^  and  ?/  =  4  as  in  equation  (2). 
Where  in  the  solution  was  one  unknown  made  to  disappear,  and  how  ? 

211 


2.    Solve 


212  A   HIGH   SCHOOL   ALGEBRA 

WRITTEN    EXERCISES 

Solze  for  X  and  y : 

1.  a;  +  ?/  =  4,  15.    2  a;  — ?/  =  0, 
Zx-\-hy  =  \%.  y-]-2x  =  ^. 

2.  x-{-y  =  l,  16.    10;?/  +  aJ  =  ll, 
3  a;  +  10  2/  =  42.  2/  +  1^  ic  =  H- 

3.  a? +  2/ =  5,  17.    Sy  — ic  =  3, 
2a; +  52/ =  22.  ^_5a;  =  -9. 

4.  a;  +  2/  =  0,  18.   4  2/  — 4  a;  =  8, 
52/  +  2a;  =  -3.  y-\-x=%. 

5.  2  a; +  2/ =  4,  19.    2  a; +  2/ =  9, 
x^2y  =  ^.  x-\-2y  =  12. 

6.  3a;  +  32/  =  0,  20.    7x-\-y  =  U, 
a;  +  4  2/  =  3.  a;  +  9  2/  =  64. 

7.  a;-2/  =  l,  21.   2  a; +  3  2/ =  22, 
2x-\-y  =  5.  5  a;  +  4  2/  =  48. 

8.  a;  — 2/  =  l,  22.    2  a; +6  2/ =  34, 
3a;  +  22/=13.  6aj  +  82/  =  62. 

9.  a;-22/  =  -4,  23.   2a;H-42/  =  38, 
2x-y  =  l.  3x-{-y  =  27. 

10.  3  a;  — 32/  =  0,  24.   x  +  10y  =  ll, 
x  +  y  =  6.  i^  +  i2/  =  TV 

11.  4  a;  — 22/  =  0,  25.    ia;  — i2/  =  6, 
x-\-Sy  =  7.  x  —  2y  =  0. 

12.  5  a; +  3  2/ =  23,  26.   5  a;  — 3  2/  =  1, 
3a;  +  52/  =  17.  2x  +  y  =  7. 

13.  5a;-32/  =  ll,  27.    2a;  +  32/  =  2, 
Sx-y  =  5.  Sx-6y  =  2. 

14.  4 a; +  2/  =  —  '^?  28.    2  a;— 3  ?/  =  4, 
2/-4a;  =  9.  7x-ey  =  3. 

293.  In  the  work  of  solving  the  preceding  problems  the 
process  has  always  been  to  combine  the  given  equations  so  as 
to  obtain  an  equation  involving  only  one  of  the  unknowns. 

The  other  unknown  is  said  to  have  been  eliminated. 


SYSTEMS  OF   LINEAR  EQUATIONS  213 

294.  Independent  Equations.  Equations  which  express  dif- 
ferent relations  between  the  same  unknowns  are  called  inde- 
pendent equations. 

Thus,  X  —  7j  =  1  and  x  -{-  y  =7  are  independent  equations  ;  for  one  ex- 
presses the  difference  between  two  numbers,  x  and  y,  while  the  other  ex- 
presses the  sum  of  the  same  two  numbers. 

X  —  y  =  1  and  2x—2y  =  2  are  not  independent,  for  the  second  when 
divided  by  2  is  the  same  as  the  first. 

295.  While  a  linear  equation  in  one  unknown  has  but  one 
root,  a  linear  equation  in  two  unknowns  is  satisfied  by  any 
number  of  sets  of  values  of  the  unknowns.  In  the  former  the 
unknown  is  a  constant,  while  in  the  latter  the  unknowns  are 
variables. 

296.  Simultaneous  Equations.  Two  or  more  equations  are 
said  to  be  simultaneous  when  all  of  them  are  satisfied  by  the 
same  values  of  the  unknowns. 

297.  Systems  of  Equations.  Two  or  more  equations  consid- 
ered together  are  called  a  system  of  equations. 

Tims,  each  of  the  exercises  of  p.  212  contains  a  system  of  equations, 

298.  A  system  of  independent  simultanous  linear  equations 
in  which  the  number  of  equations  is  the  same  as  the  number 
of  unknowns  is  satisfied  by  only  one  set  of  values  of  the 
unknowns. 

299.  Method  of  Addition  and  Subtraction.  All  systems  of 
two  independent  simultaneous  equations  of  the  first  degree  in 
two  unknown  quantities  can  be  solved  by  the  method  of  addi- 
tion and  subtraction. 

The  metJiod  of  addition  and  subtraction  consists  in  multi- 
plying one  or  both  of  the  given  equations  by  such  numbers  that 
the  coefficients  of  one  of  the  unknowns  become  equal.  Then  by 
subtraction  this  unknown  is  eliminated,  and  the  solution  is  reduced 
to  that  of  a  single  equation. 

If  the  coefficients  of  one  unknoivn  are  made  numerically  equal, 
but  have  opposite  signs,  the  equations  should  be  added. 


214 


A   HIGH   SCHOOL   ALGEBRA 


Solve: 

Multiplying  (1)  by  2, 

and  (2)  by  3, 
Adding  {3)  and  (4), 

and 
Substituting  2  for  x  in  (2), 


EXAMPLE 

6x  +  2y  =  14:. 

Sx -Qy  =  10, 

18  X  +  6  1/  =  42. 

26  x  =  52, 

x  =  2. 

2/  =  l. 


Test.     Substitute  x  =  2,  y  =  1  in  equations  (1),  (2). 


WRITTEN    EXERCISES 


Solve  and  test : 

1.  2x  +  y  =  5, 
x-y  =  l. 

2.  Sx-{-2y  =  7j 
x-2y  =  3. 

3.  2a;4-2/  =  lJ, 
x-y  =  0. 

4.  Sx-\-2y  =  2, 
x-\-y  =  l 

5.  a; +  5  2/ =  35, 
5  a; +  2/ =  31. 

6.  4a;  +  3y  =  18, 

7.  x-^2y  =  0, 
4a;-3!/  =  4. 


.8. 


10. 


2a;  +  32/  =  4, 
3x  +  2y=l. 

i^  +  i2/  =  4, 


3a;  +  .2  2/  =  .l, 
.2a;  +  .32/  =  .4. 

11.  4a;  +  ?/  =  34, 
4  2/  +  a;  =  16. 

12.  4  a;  — 2/ =  7, 
Sx  +  4:y  =  29. 

13.  2  a;  4- 3  2/ =  4, 
3a;-22/  =  -7. 

14.  2a;4-32/-8=0, 
7a;  —  2/  —  5  =  0. 


(^) 

(4) 

(^) 
(7) 


15.  Find  two  numbers  whose  sum  is  13  and  the  difference 
between  twice  the  first  and  three  times  the  second  is  3. 

16.  The  cost  of  a  house  and  lot  was  ^6500;  the  house  cost 
$  3500  more  than  the  lot.     What  was  the  cost  of  each  ? 

17.  In  a  recent  year  the  value  of  the  hay  crop  in  the  United 
States  exceeded  that  of  the  cotton  crop  by  $  30  millions ;  the 
two  amounted  to  $  1180  millions.     Find  the  value  of  each. 

18.  The  value  of  the  cotton  crop  in  a  certain  year  exceeded 
that  of  the  wheat  crop  by  $  50  millions ;  the  two  amounted  to 
$  1100  millions.     What  was  the  value  of  each  ? 


SYSTEMS  OF   LINEAR  EQUATIONS  215 

19.  If  the  value  of  the  sugar  produced,  in  a  given  year,  had 
been  increased  by  2*5-  of  itself,  it  would  have  equaled  the  value 
of  the  barley  crop ;  the  two  amounted  to  $  108,000,000.  What 
was  the  value  of  each? 

300.  Method  of  Substitution.  The  method  of  addition  and 
subtraction  can  be  used  to  solve  all  systems  of  two  simultane- 
ous equations  with  two  unknowns,  but  occasionally  problems 
occur  in  which  other  methods  are  shorter.  The  most  useful 
of  these  is  the  method  of  substitution. 

EXAMPLES 

1.  Solve:  (3x  +  2y  =  6,  (1) 

2x-    y  =  L  (2) 

From  equation  (^),                                     y  =  2x  —  i.  (3yL> 

Substituting  in  (i),            3  X  +  4  iC  —  8  =  6.  (4) 

,'.  7  a;  =  14,  (5) 

and  x=2.  (6> 

From  (6)  and  (3),                                y  =  0.     Verify.  .            (7) 

2.  Solve:  1'"-'^  =  ?'  ^'^ 

1  5  a- =  15.  (^) 

From  equation  (,^),  X  =  3.  *  (3) 

Substituting  in  (I),  12  —  3?/  =  9.  (^) 

.-.  y  =  l.     Verify.  (5) 

301.  The  method  of  substitution  consists  in  expressing  one  un- 
known in  terms  of  the  other  by  means  of  one  equation  and  sub- 
stituting this  value  in  the  other  equation,  thus  eliminating  one 
of  the  unknowns. 

This  may  be  the  shorter  method  when  an  unknown  in  either 
equation  has  the  coefficient  0,  4- 1,  or  —  1. 


WRITTEN    EXER 

DISE 

S 

Solve  by  substitution : 

1.   »  +  2/  =  75, 

3.    30^  +  2?/  = 

12, 

5.   a; +  3  2/ =11, 

3a;-3.?/  =  15. 

x-h2/  =  5. 

3aj  +  2/  =  9. 

2.    5a^  +  22/  =  31, 

4.    x-\-2y  =  l 

6.   ic  — 42/  =  8, 

x^l2y. 

x  =  iy. 

i»  +  2  2/  =  14. 

15 

216  A   HIGH  SCHOOL   ALGEBRA 

7.  7j-\-15x  =  53,  9.   3x-5y  =  81,       11.    x  +  y  =  12, 
x-\-Sy  =  27.  Ay  =  -10x.  Sx  +  y^24:. 

8.  5x-7  2/=— 35i     10.   x-\-y  =  i,  12.   x-i-2y  =  10, 
2x  —  y  =  j:                    x  +  7y  =  ^^-.  5x  —  2y  =  2. 

Solve  and  test : 

13.    x  +  y  =  29,  14.    x-\-y  =  4:S0,  15.    x  =  4.y-[-76, 

2a;  +  o2/  =  103.  12  a;  +  20  2/ =  7520.  x-y=:^30. 

Show  that  each  of  the  following  exercises  can  be  solved  by- 
using  one  equation  with  one  unknown,  or  by  using  two  equa- 
tions with  two  unknowns : 

16.  The  continued  height  of  a  tower  and  flagstaff  is  100  ft. ; 
the  height  of  the  tower  is  60  ft.  more  than  the  length  of  the 
staff ;  find  the  height  of  each. 

17.  A  house  and  lot  are  worth  $3500;  the  house  is  worth 
$2500  more  than  the  lot;  find  the  value  of  each. 

18.  The  area  of  the  United  States  and  the  British  Isles 
together  is  3,146,600  sq.  mi. ;  the  area  of  the  United  States 
diminished  by  600  sq.  mi.  is  25  times  that  of  the  British 
Isles.     What  is  the  area  of  each  ? 

19.  Japan's  exports  in  a  recent  year  plus  its  exports  10  yr. 
ago  were  approximately  $180,000,000;  this  was  a  gain  of 
400  %  on  their  value  10  yr.  before.  What  was  their  value 
at  that  time? 

20.  It  is  estimated  that  the  part  of  the  population  of  the 
United  States  living  on  farms  is  J  of  the  rest  of  the  popula- 
tion. Taking  the  total  to  be  93  millions,  how  many  live  on 
farms  ? 

21.  The'  average  creamery  of  the  Eastern  States  produces 
only  f  as  much  butter  as  the  average  creamery  of  the  West- 
ern States ;  two  average  eastern  creameries  and  three  average 
western  creameries  together  produce  42,000  lb.  annually.  What 
is  the  annual  output  of  each  ? 


SYSTEMS  OF   LINEAR  EQUATIONS  217 

302.  General  Form  of  Linear  Equations  with  Two  Un- 
knowns. A  general  form  for  an  equation  of  the  first  degree 
with  two  unknowns  is 

ax-\-by  =  e. 

A  general  form  for  two  such  equations  is 

ax-\-by  =  e  (1) 

cx-\-cly=f  (2) 

303.  General  Solution.  From  the  general  equations  (1) 
and  (2)  it  is  possible  (without  knowing  the  values  of  a,  b,  c,  d, 
e,  f)  to  find  a  general  form  for  the  solution. 

Solve:                         lax  +  by  =  e,  {!) 

cx  +  dy=f.  (2) 


Multiplying  (2)  by  c  and  {2)  by  «, 

cax 
acx 

+  cby  =  ce, 
+  ady  =  af. 

(3) 
(4) 

Subtracting  {3)  fiom  {!+), 

ady 

—  cby  =  af—  ee. 

(5) 

Thus, 

(ad- 

-  be)y  =  af-  ce. 

(0 

Dividing  by  the  coefficient  of  y. 

ad -be 

(7) 

Eliminating  by  multiplying  (I) 
by  d,  and  {2)  by  &,  and  sub- 
tracting, 

ad-  be 

(«?) 

From  these  values  of  x  and  y  the  values  in  any  particular 
case  can  be  found  by  substituting  for  a,  b,  c,  d,  e,  f  their 
particular  values.  But  there  are  other  forms  of  these  values 
better  adapted  to  substitution,  and  these  are  given  in  Chapter 
XXV. 

304.  Literal  Equations.  The  general  solution  of  Section  303 
shows  that  the  processes  of  elimination  previously  explained 
serve  fully  to  solve  systems  of  simultaneous  equations  with 
literal  coefficients. 

WRITTEN     EXERCISES 

Solve : 

1.  12  aj-H  6?/ =  -18,  3.    2x-3y  =  5a-by 
48  a:  -  9  2/  =  60.  3  x-2  y  =  b -\-5  a. 

2.  a; -1-17  2/ =  300,  4.    26  a;  +  42  2/ =  33, 
11  a;  -  2/  =  104.  39  a;  -h  28  2/  =  44. 


218  A  HIGH   SCHOOL   ALGEBRA 

b.    3x-5y  =  -l,  10.    3 p-\-q  =  3, 
6x-\-2y  =  3^.  5p-g  =  13. 

6.  12x-{-6y  =  13,  11.   4/4-26  =  1, 
^Sx-9y  =  30.  3h-\-b  =  5. 

7.  5  x-\-4:5y  =  145,  12.    4:  v  —  5  iv  =  6, 
15x-9y  =  3.  -3v  +  4:w  =  l. 

8    '^  +  -^  =  1  13.    14i?-h3r  =  2, 

a     b       '  9B-\-2r  =  6. 

_x y  _A  ^^-    (c('-\-G)x— (a  — c)y=2ab, 

2~a~3~b'~    '  {a  +  b)x-{a-b)y  =  2ac. 

9.    Solve  for  a  and  Z:  15.    {a-\-V)x— (b  ■\-l)y  =  'c, 
l=za-[-(n  —  l)d.  {a  —  l)x-\-{b  —  l)y  =  d. 

s  =  '^(a  +  T).  16.    {m-n)x+{m-\-n)y=mn, 

^  {m—]))x-{-(iin-\-p)y=mp. 

17.  An  investor  purchases  two  kinds  of  securities ;  one  kind 
pays  2  %  and  the  other  4  <fo  annually  ;  his  annual  income  from 
both  sources  is  $  900 ;  if  he  had  invested  as  much  in  the  2  % 
securities  as  he  did  in  the  4  per  cents,  and  vice  versa,  his  in- 
come would  have  been  $  600.     Find  his  investment  in  each. 


Solution.     1.   Let  x  and  ?/ be  the  number  of  dollars  invested  at  2% 

d  4%,  respectively. 

2.   Then,  by  the  conditions, 

.02  X  +  .04  y  =  900, 

3.  and 

.04:  X -\-  .02  y  =  600. 

4.   Multiplying  (2)  by  2, 

.04  x+  .OSy  =  1800. 

5.    Subtracting  (3)  from  (4), 

.06  y  =  1200. 

6.   Solving  (5), 

y  =  20,000. 

7.    Substituting  y  =  20,000  in 

(1) 

,  .02  a;  +  800  =  900. 

8.    Solving  (7), 

X  =  5000. 

.-.  he  invested  $  5000  at  2  %  and  $  20,000  at  4%. 
Test.  2  %  of  $  5000  +  4%  of  ^  20,000  =  $  900  ; 

2%  of  $  20,000  +  4%  of  $  500  =  $600. 

18.  An  investor  purchased  Pennsylvania  Railroad  stock 
paying  6  %  annually,  and  municipal  bonds  paying  4  %  ;  his 
annual  income  from  both  was  $2100;  if  the  stock  had  paid 
1  %  less  and  the  bonds  1  %  more,  his  total  income  would  have 
been  $  2000.     How  much  did  he  invest  in  each  ? 


SYSTEMS   OF   LINEAR   EQUATIONS 


219 


19.  An  investment  of  $20,000  in  one  stock,  and  one  of 
$  10,000  in  another,  together  yield  $  1300  annually  ;  an  invest- 
ment of  half  as  much  in  the  first  and  twice  as  much  in  the 
second  would  together  yield  $1100  annually.  What  is  the 
annual  rate  of  dividend  in  each  stock  ? 

20.  A  standard  daily  ration  for  an  adult  laborer  requires 
4  oz.  of  protein  and  4  oz.  of  fat. 

The  following  table  shows  the  approximate  amounts  of  pro- 
tein and  fat  in  various  foods : 


Food 

Pkr  Cent  of  Fat 

Pee  Cent  of  Protein 

Mutton 

37 

14 

Pork  (fresh) 

26 

13 

Eggs 

9 

13 

Bread  (white) 

1 

9 

Beans  (dried) 

2 

22 

Corn  (green) 

1 

3 

Rice 

I 

8 

How  many  ounces  of  mutton  and  bread  are  needed  to  make 
a  standard  ration  for  one  day  ? 

Solution.     1.    Let  x  and  y  be  the  numbers  of  ounces  required  of 
mutton  and  bread,  respectively. 

2.  Then  0.14  x  +  0.09  y  is  the  amount  of  protein   in  these   foods 
according  to  the  table. 

3.  Also  0.37  X  -\-0.01y  is  the  amount  of  fat  in  these  foods. 

4.  Thus,  0.14  x  + 0.09?/ =  4, 

5.  and  0.37  x  +  0.01  y  =  4. 

6.  Multiplying  (4)  by  100,         Ux-\-9y  =  400. 

7.  Multiplying  (5)  by  100,  37  x  +  ?/  =  400. 

8.  Subtracting  (7)  from  (6),     2Sx-Sy  =  0. 

37  .  i 


inV- 


Using  (8)  in  (7), 


23 


y  =  400. 


10.  .-.  ?/ =  28.8  (to  one  decimal  place)  and  cc  =  10.0+ 

11.  .-.  the  ration  is  10  oz.  of  mutton  and  28.8  oz.  of  bread. 

A  negative  result  for  either  unknown  quantity  would  show  that  it  is 
impossible  to  make  up  the  standard  ration  out  of  the  foods  named. 


220 


A   HIGH   SCHOOL   ALGEBRA 


Find  which  of  the  following  combinations  of  foods  can 
make  a  standard  ration,  and  the  number  of  ounces  of  each 
food  required : 


21.  Mutton  and  beans. 

22.  Mutton  and  rice. 

23.  Bread  and  eggs. 

24.  Pork  and  beans. 

25.  Eggs  and  corn. 


26.  Mutton  and  corn. 

27.  Bread  and  pork. 

28.  Bread  and  rice. 

29.  Pork  and  rice. 

30.  Eggs  and  rice. 


31.  A  man  has  a  certain  sum  of  money  invested  at  5  %  ; 
he  reinvests  the  whole  sum,  placing  three  times  as  much  of  it 
at  8  %  as  he  does  at  4  %.  His  income  is  increased  $200  a 
year  by  the  change.  How  much  money  has  he,  and  what  is 
his  income  ? 

305.  Solving  Fractional  Equations.  I.  In  case  of  fractional 
equations,  ivhen  the  unknown  quantities  occur  only  in  monomial 
denominators,  it  is  best  not  to  clear  of  fractions. 


1.    Solve: 


Multiplying  (2)  by  4, 

Simplifying  (3), 

Subtracting  (It)  from  (1), 

Dividing  (5)  by  13, 
Clearing  (6)  of  fractions, 


EXAMPLES 

2  +  ^  =  2. 
X      y 

115 


2x 


32/ 
j4 4^ 

X     Sy 
13 


'36 

20 
36* 
5 


Substituting  3  for  y  in  (i) 

Test.  -  +  -  =  2  and 


2+3 
2^3 


1 
2  •  2 


13 
Sy       9 
J_^l 
Sy      9* 
Sy  =  9. 

.'.y  =  s. 

x  =  2. 
1    ^  5 
3-3     36' 


(2) 
(5) 
(4) 

(7) 
(5) 


It  is  evident  that  the  above  equations,  if  cleared  of  fractions,  would 
contain  terms  in  xy  which  would  complicate  the  solution. 


SYSTEMS   OF  LINEAR  EQUATIONS  221 

2.   In  such  equations  the  unknown  quantities  may  be  thought 

of  as  -  and  -  • 
^        y 

Thus,  to  solve  —  +  -  =  5,  and  '^  -  ^  =  5,  is  to  solve 
X      y  X      y 

10(l)  +  9(i)  =  5,and35(l)-6(l).5. 

The  coefficients  are  now  manipulated  as  in  the  case  of  integral  equa- 
tions. It  is  often  convenient  to  introduce  new  unknowns  in  such 
problems  ;  in  this  case,  for  example,  by  putting 

~  =  x'  and  ~=v'  ^^^  equations  become  lOx'  +  9y'  —  5, 

X  y      ^ '  35  ic'  -  6  ?/'  =  5. 

306.  The  same  principles  apply  in  exactly  the  same  way  to 
equations  with  literal  coefficients. 


{2) 


EXAMPLE 

■^+^=™, 

1.    Solve: 

X     y 
X      y 

Eegard  i, 

X 

l-as 

y 

the  unknowns,  multiply  (1)  by  d  and 

Then, 

9A-^=md-nh, 

X         X 

or, 

{ad  —  be)  -  =  md  -  nb. 

Hence, 

1     md  —  nb 
X      ad  —  be 

Similarly,  multiply  (1)  by  c  and  (2)  by  a  and  subtract. 

Then, 

{ad  —  bc)-=an  —  mc. 

and 

1  _an  —  mc 

y      ad—  be 

From  (5), 

^  _  ad-  be 
md  —  bn 

From  (A), 

ad  -be 
an  —mc 

{3) 

(A) 

(.5) 
{6) 

Test.     In  literal  equations  it  is  generally  more  convenient  to  test  by 
reworking  in  a  different  way,  when  possible,  than  by  substitution. 


222  A   HIGH  SCHOOL  ALGEBRA 

WRITTEN    EXERCISES 


Solve  for  x  and  y : 
1. 


+  2"  10      12 


113 


3       2  111 

6 


^^^      22/  =  -13.  a?     2/     4 


11.  5  +  5  =  3, 

6  X     y 

-  +  ^2/  =  18,  12_20^_3 

8  .  a;       2/  ' 


--y 

X 


3x     2y         6 


4- 1+¥=-'^'  i+?=-i. 

a;     y 


X      3 


3     2a 


7/4  13. =  &, 

-^  X       y 

a^^l,  2a;       2/ 

2/      4  14.    2a;  +  -  =  14, 

y 

1(2/  —  3  x)  +  ^(9  a;  —  3  2/)  =  12.     suggestion.    Regard  x  and  -  as 

the  unknowns. 

15.    ^-2y  =  6, 

X 
X 


8. 


20       4 
X      32/ 

0, 

2     5        • 

^  +  1=1, 

.V 

a; 

3 
2 

16.    5  +  32/  =  17, 

a.' 

?-22/  =  -6. 


SYSTEMS  OF   LINEAR   EQUATIONS 


223 


X 

a 

y 

=  2, 

b__ 

X 

1 

y 

=  a. 

2a 

36 

X 

y 

Sa 

4^_ 

X 

y 

17. 


18.   ^_:i^  =  4 


2  c. 


19. 


20. 


3 

2 

+  -  = 

— 

12, 

X 

y 

4 

_3_ 

1. 

a; 

y 

1 

+^  = 

1, 

iC 

2/ 

a 

+i= 

1. 

X 

y 

21.  The  sum  of  the  reciprocals  of  two  numbers  is  |,  and  the 
difference  of  the  reciprocals  is  \.     Find  the  numbers. 

22.  Find  two  numbers  such  that  3  times  the  reciprocal  of 
the  first  added  to  5  times  the  reciprocal  of  the  second  makes 
2,  and  24  times  the  reciprocal  of  the  first  diminished  by  10 
times  the  reciprocal  of  the  second  makes  1. 

307.  II.  It  is  usually  best  to  clear  of  fractions  when  the  un- 
knowns occur  in  polynomial  denominators. 


EXAMPLES 


1.    Solve  for  x  and  y : 


Clearing  (1)  effractions, 
Clearing  {2)  of  fractions, 
Solving  {3)  and  (4), 


fa;  +  l^l 
+  1      2' 
-1      1 


[y 


1     4 


4a;  —  4  =  ?/—  1,  or4x  —  y 
X  =  2  and  y  =  ^. 


=  -  1. 


{2) 

{3) 
{4) 
(5) 


2.    Solve: 


Clearing  (1)  of 

fractions, 
Collecting  terms 

in  (3), 
From  (^), 
'Solving  (4)  and  (5), 


{2x  +  l_2y  +  l^ 
x  —  4.         2/+6   ' 

w 

x  +  1^     2. 

y 

w 

2xy  +  y+nx-^Q=2xy-\-\%x-2>y- 

-72.     (5) 

-  G  X  +  9  ?/  =  -  78. 

(4) 

x  +  2y=-\. 

(^) 

x  =  1,  y  =—4. 


(6) 


224 


A  HIGH   SCHOOL   ALGEBKA 


3.   Solve; 


Separating  the 
fractions  of  (1)» 

Simplifying  the 
terms  of  (3), 

Hence. 
Or, 

Substituting 
y=-8in(«), 


'     =y+        '    , 

X                                X 

W 

x  +  4r._     2 

(2) 

XX                     XX 

(3j 

2  +  ^  =  ^  +  10  +  ^. 

X                               X 

i4) 

2  =  y  +  10. 

(5) 

y=-s. 

(e) 

x  =  6. 


(7) 


WRITTEN     EXERCISES 


Solve : 
1. 


2. 


7. 


9. 


10. 


-2. 


2^+3l^l 
1  +  42/ 
4  a;  —  2y 
^'-2/ 

~2/  +  2' 

3^^1 

4a;-22/      2* 


3. 


4. 


a;+l 


=  4, 


^±1  =  3. 


2/ 
2a; 


c-2y 
a  +  6  — 2a;   h 


22/- 


2/  +  1 
27a;-3 
9a;     ' 


5 
? 
16 
'13' 
1 
''        3" 
c-2y 
b-2x 


.^  =  1, 


a;H-a    a 


y    ^ 

a;  — a 


2  +  1  =  2,     5_2  =  5. 
a;     2/  ^2/ 


5. 


6. 


a;-f-5      a;+l 


2/-2' 
3. 


2/-3 

a;  +  2 
2/-3 

^ill  =  _l 
02/ 

3a;  +  22/_ 


2/  +  6 

2/   _  a;  _  30  a; 

2^~3 

^  =  -1. 


10  a; 


=  2. 

52/ 


2a; 
3 


11.    ^-^ 


5  2/ 
12 


3^ 
2  ■ 


42/ 
"3 


2     a;-2/^l. 
3'    a;  4-  2/      ^ 


12.    3a;-5av=:8a5, -^-72/  +  3?>  =  0. 


SYSTEMS  OF  LINEAR  EQUATIONS  225 

14     ^±1  =  4  15     3a;  +  12y^3       ^^    4(3a;-2y)^  8 

x-y       '  '        l-\-x  '  '     5(x-\-2y)       10' 

.^    ^     4  1  +  41/  4  3a;^5 

TYPES  OF  LINEAR  SYSTEMS 

308.  Certain  systems  of  equations  have  more  than  one 
solution. 

EXAMPLES 

1.  Solve:  Sx-\-y  =  5. 

We  may  express  x  in  terms  of  y,  or  express  y  in  terms  of  x,  but 
neither  is  eliminated.  To  obtain  sets  of  numerical  values  for  both  of 
them  we  may  give  arbitrary  values  to  one,  and  find  the  corresponding 
values  for  the  other.    In  the  above  equation,  if  we  give  x  the  values, 

—  2,   —  1,  0,  1,       2  ;  and  so  on. 
y  will  have  the  values,  11,       8,  5,  2,  —  1 ;  and  so  on. 

2.  Solve:  f^  +  2/  =  3, 

[2x-\-z  =  5. 

This  is  a  system  of  two  equations  with  three  unknowns.  We  can  dis- 
pose of  only  one  of  them  by  the  methods  of  elimination,  hence  the  result 
is  an  equation  in  two  unknowns  which  may  have  any  number  of  solutions 
as  in  the  case  of  Exercise  1.  Thus,  the  given  system  has  an  unlimited 
number  of  solutions. 

309.  Indeterminate  Systems.  Systems  of  equations  like 
those  in  Section  308,  admitting  an  unlimited  number  of  solu- 
tions, are  called  indeterminate  systems. 

310.  The  equations  of  a  system  may  be  dependent. 

EXAMPLE 

Solve:  1"^-^^==^^ 

[3x-6y  =  S. 

If  the  first  equation  is  multiplied  by  3,  the  result  is  the  second  equation. 
Hence,  nothing  is  gained  by  attempting  to  eliminate  one  of  the  unknowns. 


226  A  HIGH  SCHOOL  ALGEBRA 

311.  Dependent  Equations.  If,  in  a  system  of  equations, 
two  or  more  equations  express  the  same  relation  between  the 
unknowns,  the  equations  are  said  to  be  dependent. 

In  such  a  case,  if  the  number  of  unknowns  is  the  same  as 
the  number  of  equations,  the  system  is  indeterminate. 


312.  The  equations  of  a  system  may  be  inconsistent. 

EXAMPLE 

Solve:  |^  +  22/  =  3, 

l22/  +  a;  =  4. 

The  first  equation  asserts  that  a;  +  2  y  =  3,  but  the  second  equation 
asserts  that  x  +  2  y  =  4.  Evidently  both  of  these  relations  cannot  be 
true  for  the  same  values  of  x  and  y.  If  equation  (1)  is  subtracted 
from  equation  (2),  the  result,  0  =  1,  shows  that  they  are  incompatible. 

313.  Inconsistent  Equations.  Equations  which  express  con- 
tradictory relations  between  the  unknowns  are  called  incon- 
siste7itf  or  incompatible,  equations. 

314.  Number  of  Solutions.  Two  linear  equations  in  two 
unknowns  may  be: 

1.  Determinate  and  have  one  solution. 

2.  Dependent,  and  have  an  unlimited  number  of  solutions. 

3.  Contradictory  and  have  no  solution. 

ORAL   EXERCISES 

Classify  the  following  systems  according  to  Sees.  308-313  : 

^     (x-\-y  =  2,  ^     (3x-y  =  2,  ^     iax-by  =  c, 

[x-\-y  —  z  =  3.  [9  x  —  3y  =  6.  [ax  — by  =  be. 


|2aj-2/  =  4,  (2x-y  =  7, 


2.  ^  ' 

[4.x-2y  =  12. 


6. 


[x^y=^o. 


'5  x  —  y  —  4:, 

10x-8  =  2y- 


INTERPRETATION 

315.  Problems  may  lead  to  equations  whose  solutions  are 
inconsistent  with  the  given  conditions.  Hence,  any  solutions 
that  do  not  admit  of  interpretation  should  be  rejected. 


SYSTEMS  OF   LINEAR  EQUATIONS  227 

EXAMPLES 

1.  A  shelf  contained  20  books;  some  were  histories  and  the 
rest  biographies.  If  4  times  the  number  of  biographies,  less  2 
times  the  number  of  histories,  was  35,  how  many  were  there 
of  each  kind  ? 

Solution.  a;  +  y  =  20,  (1) 

Multiplying  (i)  by  2  and  adding,         6  ?/  =  75. 

Therefore,  y  =  12^,   and  X  =  7^. 

Discussion.  The  equations  are  correctly  written  and  solved,  but  the 
values  show  that  the  conditions  of  the  problem  are  impossible,  because  we 
cannot  have  a  fractional  number  of  books. 

2.  If  the  sum  of  the  length  and  width  of  a  rectangular 
garden  is  5  ft.  and  the  difference  between  the  dimensions  is  19 
ft.,  find  the  length  and  width  of  the  garden. 

Solution.  x  -\-  y  =  6,  (1) 

x-y  =  l9.  {2) 

Adding  (I)  and  {2),  2  cc  =  24,  and  X  =  12.  (5) 

Substituting  £b  in  (^),  y  =—1 .  (^) 

Discussion.  The  equations  are  correctly  written  and  solved ;  hence, 
the  statement  of  the  problem  is  at  fault,  because  an  actual  garden  could 
not  have  a  side  measuring  —  7  ft. 

WRITTEN    EXERCISES 

1.  If  the  sum  of  two  consecutive  even  numbers  is  26,  and 
one  of  them  is  -f  of  the  other,  find  the  numbers. 

2.  If  a  positive  fraction  is  equal  to  ^,  and  the  numerator 
exceeds  the  denominator  by  5,  find  the  fraction. 

3.  If  the  sum  of  two  integers  is  24,  and  one  of  them  is  f  of 
the  other,  find  the  numbers. 

4.  The  difference  between  two  numbers  is  2,  and  one  of  them 
is  10.     Find  the  other. 

5.  If  there  are  two  numbers  such  that  the  first  plus  3  times 
the  second  equals  8,  and  twice  the  first  plus  6  times  the  second 
equals  15,  find  the  numbers. 


228  A  HIGH   SCHOOL  ALGEBRA 

EQUATIONS  WITH  THREE  OR  MORE  UNKNOWNS 

316.  The  definitions  and  methods  given  for  the  solution  of 
two  equations  with  two  unknowns  may  be  applied  equally 
well  to  a  greater  number  of  equations  and  unknowns. 

To  solve  three  linear  equations  with  three  unknowns,  eliminate 
one  unknown  from  any  pair  of  the  equations  and  the  same  unknown 
from  any  other  pair ;  two  equations  are  thus  formed  lohich  involve 
only  two  unknowns  and  which  may  be  solved  by  methods  previously 
given. 

Four  or  more  equations  with  four  or  more  unknowns  may  be 
solved  similarly. 

EXAMPLE 

p-22/  +  30  =  2,  .       (i) 

Solve:  |2ic-32/  +  «  =  l,  {2) 

13  a; -2/ +  22  =  9.  {S) 

'     To  eliminate  x  from  (i)  and  {2)  : 

Multiplying  (I)  by  2,  2  X  —  4  ?/  +  6  3!  =  4.  {4) 

Subtracting  (^)  from  (4),  —  ?/  +  5  0  =  3.  (5) 

To  eliminate  x  from  (i)  and  (3)  : 

Multiplying  (J)  by  3,  3  X  —  6  y  +  9  2;  =  6.  {6) 

Subtracting  (3)  from  {6),  — 5y  +  72;=—  3.  (7) 


To  eliminate  y  from  (,5)  and  (7)  : 


{8) 

(.9) 

(10) 

ill) 
{12) 
il3) 
iU) 


Since  x  was  found  in  step  (13)  by  substituting  y  =  2,  ^  =  1  in  equation 
(i),  it  is  not  necessary  to  substitute  again  in  this  equation  when  testing 
the  results. 


Multiplying  (5)  by  5, 
Subtracting  (7)  from  (8), 

5y-\-26z-. 
lSz  = 

.-.  z- 

=  15. 
=  18. 
=  1. 

To  eliminate  z  from  (8)  : 

Substituting  s  =  1  in  (8), 
Solving  (11), 

Substituting  y  =  2,  «  =  1  in 
Solving  (13), 

a), 

-  5 1/  +  25  = 

y-- 

x-4  +  3z 
X-- 

=  15. 
=  2. 
=  2. 
=  3. 

Test.     (2)  2.3-3.2+1 

(5)  3  .  3  -  1  .  2  +  2 

•  1 
.1 

=  1. 

=  9. 

SYSTEMS   OF   LINEAR   EQUATIONS        *         229 

317.   Literal  equations  are  solved  in  the  same  way;   when 
fractions  are  involved  use  the  method  of  Sections  305  and  307. 

WRITTEN   EXERCISES 


Solve : 

1. 

2  a;  +  3  2/ -1- 4  2 : 

=  20, 

11. 

1  x  +  13y  =  205, 

Sx-\-4.y-\-5z: 

=  26, 

14  a;  4-  5  2J  =  300, 

3x-\-5y-\-6z-. 

=  31. 

12  2/  H-  20  2  =  140. 

2. 

x-^y-\-z  =  5, 
x  +  y-z  =  7, 
x-y-z  =  S. 

12. 

M  +  ^31, 
1  +  1^-1  =  23.5, 

3. 

x  +  2y  =  7, 

y  +  2z  =  2, 
Sx-\-2y  =  z~ 

1.- 

13. 

1  +  ^  +  1  =  19. 
4     o     6 

x-{-z  =  l  —  cy. 

4. 

5x-\-3y  =  65, 
2y-z  =  ll, 
3  a;  +  4  z  =  57. 

14. 

•                                       O  7 

y-{-z=—cx, 
x  +  y  =  —  l  —  cz. 

5. 

y  +  z  =  --a, 
x  +  z=-b, 
x-\-y=  —  c. 

3' 

6. 

x-\-y  —  z  =  l, 

.=s+|- 

^x  +  3y-^z-- 

=  1, 

4.x  +  y  —  3z  = 

1. 

15. 

X     y      z 

7. 

x-{-y-\-2z==2 

(b+c), 

2,3,4 

x+2y+z=2 

(a  +  c). 

-  +  -  +  -=  -3, 
X     y      z 

2x+y+z=2 

(a  +  b). 

?-^-5=u. 

8. 

^x-\-iy=:12- 

-4^, 

X      y      z 

iy+iz  =  S-{-ix, 

Suggestion.     First      solve      for 

ix-^iz  =  10. 

1     1 

1 

9. 

x  +  ay  =  b, 
ax-\-z  =  Cy 
z  -^cy  =  a. 

16. 

z 

ax-\-by  =  1, 
cy  —  az  =  1, 
bz  —  ex  =  1. 

10. 

x-\-^y  =  100, 

Suggestion.     Multiply  the  equa- 

y +  iz  =  100, 

tions  1 

oy  c,  b,  a,  respectively,  and 

z-\-^x  =  100. 

add. 

230  A  HIGH  SCHOOL  ALGEBRA 

17.  A  man  has  three  sums  invested,  the  first  at  4%,  the 
second  at  5%,  and  the  third  at  6%  per  annum.  The  annual 
yield  of  the  first  and  second  sums  together  is  $  220,  that  of  the 
first  and  third  together  is  $420,  and  that  of  the  second  and 
third  is  $400.     How  many  dollars  are  there  in  each  sum  ? 

18.  A  tourist  spent  $520  on  a  trip.  If  he  had  cut  down  his 
transportation  expenses  ^,  his  hotel  bill  \,  and  his  miscel- 
laneous expenses  |-,  his  trip  would  have  cost  him  $  350.  If  he 
had  cut  down  his  transportation  expenses  ^,  increased  his  hotel 
bills  by  ^,  and  his  miscellaneous  expenses  by  J,  the  trip  would 
have  cost  him  $  535.  Find  the  amount  he  actually  spent  for 
each  of  the  three  items. 

19.  Find  three  numbers  such  that  the  difference  between 
the  reciprocals  of  the  first  and  second  is  ^,  between  the  recipro- 
cals of  the  first  and  third  is  J,  and  the  sum  of  the  reciprocals 
of  the  second  and  third  is  j\. 

20.  The  sum  of  the  reciprocals  of  three  numbers  is  -^^ ;  the 
difference  between  the  reciprocals  of  the  first  and  second 
equals  that  between  the  reciprocals  of  the  second  and  third. 
The  third  number  is  twice  the  first.     Find  the  numbers. 

21.  Three  brothers,  A,  B,  C,  at  a  family  reunion  were  dis- 
cussing their  ages.  C  said  to  A,  "  Thirty  years  ago  my  age 
was  double  yours."  Then  B  said  to  A,  "  Twenty-three  years 
ago  7ny  age  was  double  yours."  If  C's  present  age  exceeds 
A's  by  four  years,  and  B's  exceeds  A's  by  eleven  years,  find 
the  age  of  each. 

318.  Equations  with  More  than  Three  Unknowns.  The  solu- 
tion of  a  particular  example  will  serve  to  indicate  the  method 
to  be  used  when  there  are  more  than  three  unknowns. 


EXAMPLE 


Solve 


2v  +  x-\-y-\-z  =  S,  (1) 

3w-2x-y-h2z  =  17,  (3) 

5^u-x-2y-3z  =  -<^.  (4) 


SYSTEMS  OF  LINEAR  EQUATIONS  231 


To  eliminate  z : 

/I 

Adding  (i)  and  (2), 

3wj  +  ix  +  fy  =  9. 

(5) 

Subtracting  twice  (i)  from  (3), 

wj-4a;-3y  =  1. 

(6') 

Adding  three  times  (i)  and  (U), 

8to  +  2x  +  ?/  =  15. 

(7') 

To  eliminate  x : 

Subtracting  (7)  from  4  times  (5), 

4^7+133^  ^21. 

{8) 

Adding  twice  (7)  to  (6), 

17  to -2/ =  31. 

(9) 

To  solve  for  to : 

Substituting  y  from  (9)  into  (§), 

4^  +  j_3(i7^o_31)=21. 

(10) 

Solving  (JO), 

233 

to  =  466,  and  w  =  2. 

(ii) 

To  solve  for  the  remaining  unknowns  : 

Substituting  w  =  2  in  (9), 

y  =  3. 

(12) 

Substituting  y  =  3,  -mj  =  2  in  (7), 

16  +  2a;  +  3  =  15. 

(13) 

Solving  (J5), 

x=-2. 

iU) 

Substituting  aj  =  -  2,  y  =  3, 
w  =  2  in  (J), 

2-2  +  3  +  ;?  =  8. 

(15) 

Solving  (15), 

;S  =  5. 

(16) 

Test,     (i)  Used  in  (15), 

{2)  2-^(-2)4-i.3 

-5: 

=  1. 

(5)  3.2-2.  (_2)-3  +  2. 

5  =  17. 

{4)  5.2-(-2)-2.3. 

-3. 

5  =-9. 

In  elementary  algebra  little  emphasis  should  be  placed  on  problems 
with  more  than  three  unknowns ;  in  later  mathematics  better  and  more 
general  methods  are  given  for  dealing  with  them,  together  with  proofs  of 
the  assumptions  which  are  tacitly  made  at  this  stage. 

WRITTEN  EXERCISES 

Solve : 

1.  v-\-w  =  Q,  3.   |x  +  32/  =  23, 
2-w;  =  8,  x  +  \z  =  ^, 
32-2^  =  10,  2/  +  32;  =  31, 

2  a;  -  ^  =  12,  a;  +-  w  +-  ?/  =  22. 

2/ +  2  =  14.  4.   w; +-0^  +  2/ =9, 

x  +  y  +  z  =  ^, 

2.  a;-22/-h32  =  32,  2/  +  ^  +  w  =  9, 
2w  — 3v+-4x  =  13,  z  +  n^-\-x  =  ^. 

"^     ^  "^         ~  ~    '  Suggestion.     Add  the  equations, 

^z  —  lw-\-5v  —  0,  divide  by  3,  and  subtract  each  from 

5  -?;  —  a;  +  3  ?/  =  —  14.  the  result. 
16 


232  A   HIGH   SCHOOL  ALGEBRA 

5.   ^w-x  +  y-2z  =  ~^,  9     1_1  =  1, 

w-Zx-2y-\-z  =  l^,  '    X     y      6' 

5w  +  2a;-32/4-22  =  19,  i_l  =  i-, 

2w-2x^-4.y-^z  =  -l^.        y     z      12' 

113 


x  —  y  +  z  =  —  ^i 

y  —  Z  +  W  =  5, 


X 


z      4 


a  ,  b 


^_^4-aj  =  _6.  10.    -  +  ^  =  ^^' 

7.  w  +  2/  +  2;  =  2aj,  b,G_ 

w  +  x  +  2  =  3y,  .             ^■^2"''' 

w  +  a;  +  2/  =  4:2,  ^  +  £  =  » 

w-x-y  =  w.  a;     ;<; 

8.  7  0^-3?/  =  1,  _ 
8.-7x  =  l,  11-    ^  +  2/  +  ^  =  l. 
4.-72/  =  l,  5a.  +  2/-.22  =  .5, 
14x-3^  =  l.  2a.  +  32/  +  3.  =  l. 

REVIEW- 
WRITTEN   EXERCISES 

Solve  and  test : 

1    ^  +  .  =  27,  7-   2(3..+l)-3(42/-26)=2, 

^:!:?  =  17.  3(.-5)  +  2(,-14)  =  6. 

_  8.   2p  +  3w;  =  l, 

2.  3.j  +  5?y  =  19,  5^7^^6. 

7a:-42/  =  13.  ^ 

9.   4  71  +  71;  =  9, 

3.  W^^)==K^  +  yl  2/i  +  5^  =  6. 

i(ll4-a5  +  2/)  =  i(9  +  2/)-  .       '  r. 

7^      -r     -r^y  10^    2  X' +  6  2/ +  7  2  =  4, 

4.  &a5  +  a2/  =  ft,  3  a;  _{_  8  ?/  +  9  2  =  —  2, 
ax  — by  =  a.  4:X-\-9  y -\-10z  —1, 

5.  a; +2/ +  2  =  4,  11.   4fl5  +  ll2/  +  2  2  =  33, 

2a7  +  32/-:2  =  l,  15a;4-392/  +  7  2  =  115, 

Sx-y  +  2z=l.  23  0^  +  56?/  + 10  2!  =  162. 

6.  ix  +  iy  =  5,  12.   2x-9y=-l, 
|a;-f2/  =  8f  5i»-242/  =  2. 


13. 


SYSTEMS  OF  LINEAR  EQUATIONS  233 

16.  6x-{-2y+Sz=l, 
.9x-^Sy  +  llz=-l, 

15x-\-12y-3z=-l, 

17.  x  —  2y-^Sz  —  u  =  5, 
14.   3a;  +  5  7  =  l,  y-2z  +  Su-x  =  0, 


X 

a 

^1= 

-7, 

X 

2 

-^1= 

:2« 

+  &. 

3 

x-\-5 

y  = 

=  1, 

4 

x-\-6 

y  = 

:3. 

X 

a 

H= 

=  1, 

X 

I 

a 

1. 

z  —  2u-\-3x  —  y  —  ()j 
u-2x  +  Sy-z  =  Q. 

^^'   n.'^h'^^'  18.    Solve  for  a  and  6 : 

ac  +  6g  =  cZ, 
ad  +  oh  =q. 

Solve  the  equations  of  Exercise  18  for : 

19.  a  and  q.  21.   c  and  q. 

20.  c  and  d.  22.    h  and  d. 

23.  -A  certain  number  is  written  with  two  digits ;  twice  the 
tens'  digit  plus  the  units'  digit  makes  9 ;  when  the  digits  are 
interchanged,  the  number  formed  is  27  greater  than  the  given 
number.     Eind  the  original  number. 

Suggestion.     1.   Let  x  =  the  tens'  digit,  and  y  =  the  units'  digit. 

2.  Then  the  number  is  10  a;  +  y,  and  10  y  +  x  is  the  number  with  digits 
interchanged. 

3.  Then,  by  the  conditions  of  the  problem, 

2x-^y  -9;  and  10  y  +  a;  —  27  =  10  a;  +  y. 

4.  Solve  these  equations  for  x  and  y. 

24.  In  a  certain  number  of  two  digits  the  sum  of  three  times 
the  tens'  digit  and  twice  the  units'  digit  is  36 ;  if  the  digits  are 
interchanged,  the  number  formed  is  27  greater  than  the  original 
number.     What  is  the  original  number  ? 

25.  The  sum  of  two  numbers  is  1.6  and  their  difference  is  .2. 
What  are  the  numbers  ? 

26.  The  difference  between  two  numbers  is  30,  and  the  less 
is  f  of  the  greater.     What  are  the  numbers  ? 

27.  Two  numbers  are  such  that  if  the  first  is  increased  by 
14,  the  result  is  twice  the  second;  if  the  second  is  diminished 
by  12,  the  result  is  ^  the  first.     What  are  the  numbers  ? 


234  A   HIGH   SCHOOL  ALGEBRA 

28.  The  angle  c  in  the  figure  is  bi- 
sected ;  and  it  is  known  that  the  bisector 
of  an  angle  divides  the  opposite  side  of 
the  triangle  into  parts  proportional  to 
the  adjacent  sides;*  what  is  the  ratio  of 
a;  to  ?/  in  the  figure  ?  What  is  the  sum 
of  X  and  2/?  Find  the  length  of  each 
^^  "^-  segment. 

29.  A  person  leaves  an  estate  worth  $13,000;  some  of  it  is 
willed  to  a  college,  and  12  times  as  much  to  an  eldest  son, 
whose  share  is  1^  times  as  much  as  that  of  each  of  his  2 
brothers,  and  If  times  that  of  each  of  5  sisters.  Find  the 
amount  left  the  college. 

30.  The  sum  of  two  numbers  is  a^  and  their  difference  is 
6^;  what  are  the  numbers? 

31.  A  and  B  play  at  a  game  with  counters.  In  the  first 
game  A  loses  as  many  counters  as  B  has ;  in  the  second  game 
B  loses  as  many  counters  as  A  then  has;  at  the  end  of  the 
second  game  A  has  16  counters  and  B  has  4.  How  many  had 
each  at  first  ? 

32.  A  certain  number  is  twice  another;  their  difference 
divided  by  their  sum  equals  the  smaller.     Find  the  numbers. 

33.  What  fraction  becomes  equal  to  |  if  the  numerator  and 
the  denominator  are  each  increased  by  1,  and  equal  to  \  if 
they  are  each  diminished  by  1? 


34.  A  fraction  whose  value  is  f  assumes  the  value  |-  if  the 
numerator  and  the  denominator  are  each  increased  by  8.  Find 
the  fraction. 

35.  A  fraction  becomes  equal  to  f,  if  the  numerator  and  the 
denominator  are  each  increased  by  3 ;  and  equal  to  \^  if  the 
numerator  and  the  denominator  are  each  diminished  by  3. 
Find  the  fraction. 


SYSTEMS   OF  LINEAR  EQUATIONS  235 

36.  The  number  128  is  the  sum  of  two  numbers  such  that 
I  of  one  equals  -^  of  the  other.     What  are  these  numbers  ? 

37.  A  certain  capital  is  invested  in  two  kinds  of  securities, 
one  paying  4%,  the  other  4^%;  f  of  the  capital  is  invested  in 
the  first  kind  and  the  rest  in  the  second ;  the  total  income  is 
$  75.     What  is  the  capital  ? 

38.  A  certain  kind  of  woolen  cloth  1  yd.  wide  shrinks  ^  of 
its  length  and  -^-^  of  its  width  in  washing.  How  many  yards 
must  be  bought  in  order  to  have  38  sq.  yd.  after  shrinking? 

39.  A  band  of  smugglers  found  a  cave,  which  would  exactly 
hold  the  cargo  of  their  boat,  namely,  13  bales  of  silk  and  33 
casks  of  rum.  While  they  were  unloading  a  revenue  cutter 
was  sighted,  and  they  sailed  away,  leaving  9  casks  and  5  bales; 
these  filled  only  one  third  of  the  cave.  How  many  bales  alone 
would  the  cave  hold  ?     How  many  casks  ? 

Suggestion.    Let  x  equal  the  number  of  bales  alone  required  to  fill  the 

cave  and  y  the  number  of  casks.     Then,  1  bale  will  fill  -  of  the  cave  and 

^  X 

1  13      33 

1  cask  will  fill  -  of  it ;  also  according  to  the  problem  —  H =  the  whole 

y  ^  ^  X       y 

capacity  of  the  cave,  or  1. 

Similarly,  form  the  equation  corresponding  to  \  of  the  capacity. 

40.  A  man  had  two  sums  invested,  one  at  4  %,  the  other 
at  5  %,  simple  interest,  and  thus  received  $  500  annually.  If  the 
rates  of  interest  had  been  5  %  and  6  %,  respectively,  he  would 
have  received  $  110  more  per  annum.     Find  the  sums  invested. 

41.  A  man  can  walk  2i  miles  an  hour  up  hill  and  3|-  miles 
an  hour  down  hill.  He  walks  bQ  miles  in  20  hours  on  a  road 
no  part  of  which  is  level.     How  much  of  it  is  up  hill  ? 

42.  Three  thousand  doHars  was  given  annually  to  a  college 
to  provide  annual  scholarships  of  grades  a,  h,  c.  When  two  of 
grade  a,  six  of  grade  b,  and  one  of  grade  c  were  granted,  the 
gift  was  just  sufficient;  similarly,  when  four  of  grade  a,  two 
of  grade  6,  and  two  of  grade  c  were  granted,  and  also  when  one 
of  grade  a,  five  of  grade  b,  and  five  of  grade  c  were  granted,  the 
gift  was  sufficient.     What  was  the  value  of  each  scholarship? 


236  A   HIGH   SCHOOL  ALGEBRA 

43.  The  number  of  adults  and  the  number  of  children  likely 
to  attend  a  certain  entertainment  was  estimated  in  advance; 
the  sum  to  be  raised  by  the  entertainment  was  $550;  if  the 
admission  for  adults  was  fixed  at  40  cents  and  that  for  chil- 
dren at  30  cents,  the  estimated  receipts  would  lack  $90 
of  the  required  amount;  but  if  the  admission  was  fixed  at 
50  cents  for  adults  and  25  cents  for  children,  the  exact 
sum  would  be  raised.  How  many  of  each  were  expected  to 
attend  ? 

44.  A  tank  contains  20  gal.  of  water,  and  water  flows  in  at 
the  rate  of  5  gal.  per  minute.  At  the  same  time  a  second  tank 
contains  50  gal.  of  water,  and  water  flows  in  at  the  rate  of  2 
gal.  per  minute.  Construct  a  graph  to  represent  the  amount 
of  water  in  the  first  tank  for  each  minute  from  0  to  15.  In 
the  same  figure  draw  a  graph  to  represent  the  amount  of 
water  in  the  second  tank  for  each  minute.  From  the  graph 
read  at  what  time  the  two  tanks  will  contain  equal  amounts 
of  water  and  what  the  amount  is.  Verify  by  solving  alge- 
braically. 

45.  A  merchant  pays  $  10  rent  weekly.  His  profits  on  his 
sales  average  20%.  Represent  graphically  his  net  profits 
corresponding  to  weekly  sales  ranging  from  0  to  $200.  A 
second  merchant  sublets  part  of  his  store  for  $  5  per  week  more 
than  his  own  rental,  but  he  makes  only  10  %  average  profits 
on  his  sales.  In  the  same  figure  represent  his  total  profits  for 
sales  ranging  from  0  to  $200.  From  the  graph  read  the 
amount  of  sales  for  which  both  merchants  make  the  same  net 
profit.     Verify  by  solving  algebraically. 

46.  If  tin  and  lead  lose,  respectively,  -^j  and  2%  of  their 
weights  when  weighed  in  water,  and  a  60-lb.  mass  of  lead  and 
tin  loses  7  lb.,  find  the  weight  of  the  tin  in  this  mass. 

47.  What  are  the  sides  of  a  rectangle  such  that:  (a)  the 
area  is  not  changed  if  the  base  is  diminished  by  2  and  the 
altitude  increased  by  2  ;  (h)  the  area  is  increased  by  10,  if  both 
base  and  altitude  are  increased  by  1  ? 


SYSTEMS   OF   LINEAR   EQUATIONS  237 

48.  The  income  from  a  certain  investment  is  devoted  to 
scholarsMps  of  two  grades ;  the  higher  grade  receives  $  200 
more  than  the  lower  per  scholarship.  When  there  are  7  stu- 
dents holding  the  lower  grade  and  7  holding  the  higher,  the 
income  exceeds  the  expense  by  $100;  but  the  income  is 
exactly  sufficient  to  provide  8  scholarships  of  the  higher  grade 
and  5  of  the  lower  grade.  Find  the  amount  of  the  income,  and 
the  amount  of  each  grade  of  scholarships. 

49.  The  sum  of  the  reciprocals  of  the  first  and  third  of 
three  numbers  is  twice  the  reciprocal  of  the  second;  the 
reciprocal  of  the  third  is  4  times  that  of  the  first ;  the  sum  of 
the  reciprocals  of  the  first  and  second  is  7.     Find  the  numbers. 

50.  A  man  has  three  debtors,  of  whom  A  and  B  together 
owe  him  60  pounds,  A  and  C  80  pounds,  and  B  and  C  92 
pounds.  How  much  did  each  one  owe  ?  (Saunderson's 
Algebra,  1740.) 

51.  A  vessel  filled  with  water  has  three  orifices.  A,  B,  G. 
If  all  three  are  opened,  it  is  emptied  in  6  hr.  ;  through  B  alone 
it  is  emptied  in  f  of  the  time  that  it  would  take  through  A 
alone  ;  and  the  time  through  G  is  twice  as  great  as  through  B. 
In  what  time  is  the  vessel  emptied  through  each  orifice  alone  ? 
(Bossut's  Algebra,  1773.) 

52.  The  price  of  a  house  is  100  dollars.  A  could  pay  for  it 
if  he  had  half  of  B's  money  in  addition  to  his  own ;  B  could 
pay  for  it  if  he  had  one  third  of  C's ;  and  C  could  pay  for  it  if 
he  had  one  fourth  of  A's  money.  How  much  had  each? 
(Euler's  Algebra,  1770.) 

SUMMARY 

The  following  questions  summarize  the  definitions  and  pro- 
cesses treated  in  this  chapter  : 

1.  What  are  independent  equations?  Sec.  294. 

2.  What  are  simultaneous  equations  ?  Sec.  296. 

3.  State  and  explain  a  method  of   solving   simultaneous 
equations.     Another  method.  Sees.  299,  301. 


238  A   HIGH   SCHOOL   ALGEBRA 

4.  When  should  the  method  of  adding  be  preferred  ? 

Sec.  299. 

5.  When  should  the  method  of  substitution  be  preferred  ? 

Sec.  301. 

6.  What  is  a  general  form  of  a  system  of  two  simultaneous 
linear  equations  ?  Sec.  302. 

7.  State  the  formulas  for  the  values  of  the  unknowns  in  a 
system  of  two  simultaneous  linear  equations.  Sec.  303. 

8.  In  solving  fractional  simultaneous  equations,  when  is  it 
best  not  to  dear  of  fractions?  When  may  the  equations  be 
cleared  of  fractions  ?  Sees.   305,  307. 

9.  When  are  equations  dependent?  Sec.  311. 

10.  When  are  two  equations  inconsistent  or  contradictory? 

Sec.  313. 

11.  What  is  meant  by  interpretation  of  results?         Sec.  315. 

12.  State  a  method  for  solving  a  system  of  three  or  more 
simultaneous  linear  equations  of  the  first  degree.  Sec.  316. 


CHAPTER   XX 
INVOLUTIO]^  AND  EVOLUTION 

INVOLUTION 

319.  The   operation   of  raising   an   expression   to  a  given 
power  is  called  involution. 

An  important  case  is  the  involution  of  a  binomial. 

320.  Preparatory. 

1.  We  have  already  found  that 

(a  H-  5)2  =  a^  +  2  a6  +  b^ 

(a  +  bf  =  a^-\-S  a'b  -^Sab^-\-  b\ 

(a  +  6)*  may  be  found  by  multiplying  the  last  result  by  a  +  6.     Thus, 
a^  +  3a^b  +  S  ab^  +  h^ 

g  +  & 

a*  +  3  a36  +  3  a^i)2  +    ct^s 

a^b  +  3  a^b^  +  3  g^^  +  54 
.-.  (a  +  &)*  =  g*  +  4  g8&  +  6  a'^b^  +  iab^  +  &* 

2.  From  this  find  similarly  (a  +  by. 

3.  From  "(a  +  bf  find  similarly  (a-\-by. 

321.  The  result  of  multiplying  out  a  power  of  a  binomial  is 
called  a  binomial  expansion. 

322.  The  coefficients  of  the  successive  powers  of  a  binomial 
may  be  arranged  in  a  triangular  table : 

Expansions  Coefficients 

(g  + 6)0  =  1  1 

(a  +  by  =  a  +  b  11 

(a  +  6)2  =  a2  +  2g6  +  &2  1    2     1 

(g  +  6)3  =  a3  +  3  a%  +  Sab^  +  b^  13    3     1 

(a  +  6)4  =  ^4  +  4  ^3^  4.  6  g262  +  4  g63  +  6*  14     6    4     1 

Supply  the  next  two  lines,  using  the  results  of  Exercises  2 
and  3,  Section  320. 

289 


240  A   HIGH   SCHOOL   ALGEBRA 

323.  Binomial  Coefficients.  The  table  on  the  right  in  Sec- 
tion 322  is  called  Pascal's  triangle,  and  the  numbers  the 
binomial  coefficients. 

Each  number  of  Pascal's  triangle  is  the  sum  of  the  number 
directly  above  it  and  the  number  to  the  left  of  that. 

That  this  must  be  so  follows  from  the  process  of  multiplying  by  «  +  &, 
and  is  readily  seen  when  the  method  of  detached  coefficients  is  used 
(Chapter  XXIV).  The  tg,ble  enables  us  easily  to  write  expansions  of 
successive  powers  of  a  -{-b. 

324.  Any  expression  that  can  be  put  into  the  form  of  a 
binomial  expansion  may  be  written  as  a  power  of  a  binomial  by 
inspection. 

For  example  : 

=  {2xy-4-Sx^y  +  6-4  xV  _  4  .  2  xy^  +  y* 
=  {2xy  +  4  .  (2a;)3(_  y)+Q(2xy(-  yY  +  4(2a;)(-  y)8  +  (-  yY 
=  {2x-yY. 
2.   a3  +  3  ^25  +  3  a52  +  53  _  3  «2c  _  6  ahc  -  3  b'^c  +  3  ac2  +  3  6c2  -  c^ 
=  (a  +  bY  +  3(a  +  b^i-  c)+  3(a  +  &)(-  c)2+(-  c)^ 
=  {a  -|-6-c)8. 

ORAL    EXERCISES 

Express  as  a  power  of  a  binomial : 
1.    _a3  4.3a2_3a  +  l. 

3.    l-3a;4  +  3a^-a;^2 

5.  Sa^-12a'b  +  6ab^-b^ 

6.  X*  —  4:  a^y  +  6  x^y^  —  4  xy^  +  y*. 

7.  a^— 4a^  +  6a;2  —  4a;  +  l. 

8.  (2xy-4.(2xy-{-6(2xy-4:(2x)  +  l, 

9.  16a^  +  32a^  +  24a;2  +  8a;4-l- 

10.  16  «4  -  32  a^  +  24  a;2  -  8  a;  +  1. 

11.  a^  -  5  a^  4- 10  a^  -  10  a2  +  5  a  -  1. 

12.  a2  +  2  a6  +  &'  -  2  ac  -  2  6c  +  c^. 


INVOLUTION   AND   EVOLUTION  241 

13.  (2  ay  +  4(2  ay  +  6(2 af  +  4(2  a)  +  1. 

14.  oc^  +  5x*y  -^  10  a^y^  +  10  xy  +  5  xy^  +  2/^- 

15.  (a  -by -4:  (a-  bfc  -{-6  (a-  bye"  -  4  (a  -  by  +  c^ 

16.  i»6  -  6  a;5  (2  ^)  +  15  a;4  (2  y)^  -  20  a^  (2  y)^  +  15  x^  (2  y)^  - 
6x{2yf  +  {2y)\ 

17.  State  in  order  the  coefficients  in  the  expansion  of  a 
binomial  of  the  fourth  degree.  Of  the  third  degree.  Of  the 
fifth  degree. 

325.  The  Binomial  Formula.     It  can  be  proved  that 

(a +  6)"  =a"+yza"-^6+^'^''~-^^  a^-W  +  '^i?^  - ^){'^^  -  2)  ^n-3^3 

n(n-l)(n-2)(7i-3)  .^^ 

^  2.3.4  ^      ■ 

This  is  known  as  the  binomial  formula. 

The  factor  1  is  understood  in  each  denominator;    and  by 

denoting, the  product  1  •  2  •  3  by  3  !  (read  "three  factorial"), 

and  generally  1  -2  -  3  -"k  by  kl,  the  above  formula  can  be 

written : 

/     .  i-\«        «  .       r.-i7.  .  n(n  —  l)a''~'^b^  ,  n(n  —  l)(n  —  2)    „_„, 
(a  +  6)"  =  a"  +  wa"  ^b-\--^ -j h-^ ^"~' — ^    ^ 

^   n(n-l)(n-2)(n-3)^^_,^,   ^  , 

4! 

326.  The  expansion  of  (a  -f  b)"  may  be  written-  by  observing 
that : 

1.  Tlie  first  term  a",  the  last  is  b",  and  the  number  of  terms  is 
n  +  1. 

2.  The  exponent  of  a  is  one  less  in  each  succeeding  term. 
b  occurs  in  the  second  term,  and  its  exponent  increases  by  one  in 
each  succeeding  term, 

3.  Tlie  coefficient  of  any  term  is  the  coefficient  of  the  next  pre- 
ceding term  multiplied  by  the  exponent  of  a  in  that  term,  and 
divided  by  one  more  than  the  exponent  of  b. 

4.  All  of  the  signs  are  plus  if  the  sign  of  b  is  plus,  or  alter- 
nately plus  and  minus  if  the  sign  of  b  is  minus. 


242  A  HIGH  SCHOOL  ALGEBRA 

WRITTEN    EXERCISES 

1.  Test  the  binomial  formula  just  given  for  n  =  S. 

It  should  reduce  to  the  known  expression  for  (a  +  6)3.  The  formula 
comes  to  an  end  because  the  factor  n  —  S  occurs  in  all  the  terms  after  a 
certain  one,  and  when  n  =  3,  this  factor  is  zero. 

2.  Test  similarly  for  w  =  4,  5,  6,  2,  1. 
Expand : 

3.  (x  +  yy.  1.    {x-  If.  11.  {ah  -  cdy. 

4.  {x-yy.  8.    {x-2yf.  12.  {t  -  uf. 

5.  (x-yy,  9.    (ab  +  iy,  13.  {3  a -{-by. 

6.  ix-{-yy.  10.    (6c -ly.  14.  {x-\-5y.' 

15.  Write  the  next  two  terms  of  the  binomial  formula  as 
given  in  Sec.  325.  ' 

16.  Observe  that  if  the  successive  terms  were  written  accord- 
ing to  the  same  law,  the  tejith  term  of  the  binomial  formula  would 
be  ^(^^  -  l)(n  -  2) . . .  (n  -  8) ^,,9^3^     ^^.^^  ^^^  fifteenth  term. 

17.  By  reference  to  the  binomial  formula,  state  the  number 
of  the  last  term  written  in  the  following  expression : 

ix^2yy  =  :^^^x\2y)^^x\2yy^  ... 

+  ^-^-^^f'^'^.^(2,)e+.... 

Write  the  first  three  terms  and  the  seventh  term  in  the  ex- 
pansion of  each  of  the  following : 

18.  (x  +  ?/)i2.  20.    (ab-yyK  22.    (2  x  -  ly. 

19.  (1  +  5  a?)".  21.    ff-{-A'''  23.    (4  0^  +  3)20. 

24.  Eeduce  the  terms  written  for  Exercise  21  to  their 
simplest  form. 

25.  If  the  expansions  in  Exercises  18-23  were  written  out 
in  full,  how  many  terms  would  each  have  ? 


INVOLUTION  AND  EVOLUTION  243 

26,  In  each  of  Exercises  18-23,  determine  whether  or  not 
there  is  a  middle  term  in  the  expansion.     If  there  is  a  middle 

term : 

(a)  Determine  its  number. 

(6)  Write  it  out,  without  simplifying. 

(c)  Simplify  the  result. 

Expand : 

28.    (a'^b  -  ^y.  30.    {x^  -  5  y^f, 

31.  Write  the  first  five  terms  of  (  w^ 

\         2  2/ 

32.  Write  the  first  five  terms  of  (2  a;  —  ^  I  . 

33.  Write  the  first  four  terms  of  (2  x^  —  y^y. 
Write  the  last  four  terms  of : 

34.  {x-'-W-  35.    U-\ 

EVOLUTION 

327.  The  process  of  extracting  an  indicated  root  is  called 
evolution.  The  most  important  case  of  evolution  is  the  ex- 
traction of  square  root. 

All  the  numbers  that  we  have  hitherto  considered,  whether  positive  or 
negative,  have  positive  squares ;  none  of  them  has  a  negative  square, 
consequently  the  square  root  of  a  negative  number  (as,  for  example, 
V—  5)  has  no  meaning  at  this  stage  of  our  work,  but  will  be  explained 
in  Chapter  XXVIII. 

328.  Square  Root  by  Inspection.     The  formula 

[±(a-f-&)]'  =  a'4-2a6  +  &2^ 

shows  that  when  one  of  the  terms  of  the  trinomial  is  twice  the 
product  of  the  square  roots  of  the  other  two,  the  trinomial  is 
the  square  of  the  sum  of  these  square  roots.  By  aid  of  this 
relation  the  square  roots  of  certain  trinomials  can  be  found 
readily  by  inspection. 


244  A  HIGH   SCHOOL   ALGEBRA 

For  example : 


\/4:  a'^  —  12  ab  +  9  b'^  =  ±  (2  a  -  3  b),  since  —  12  ab  is  twice  the  product 
of  the  square  roots  of  4  a^  and  9  b^.  These  roots  must  be  taken  with  oppo- 
site signs  in  this  case  because  twice  their  product  is  to  be  negative. 

ORAL    EXERCISES 

Find  the  square  root  of : 


1. 

a;2  +  4a;-f-4. 

8. 

4a252_4a&  +  l. 

2. 

4a;2-f8a;  +  4. 

9. 

3. 

a262  +  2a6  +  l. 

10. 

x^--^2x^''i/-^y^. 

4. 

a2_4a?>-f-4  62. 

11. 

25a''-10a'^l. 

5. 

4  7?i2  _  4  ^i^i  _j_  ^^2_ 

12. 

25  a^  -  30  ab-j- 9  h\ 

6. 

a;4_4x2_^4. 

13. 

^<daW-Uo?h  +  a\ 

7. 

16a^  +  8x4  +  l. 

14. 

25  a4&2^2  _^  10  ^25^5  _f_  c? 

329.  Square  Roots  of  Arithmetical  Numbers.  The  square 
root  of  arithmetical  numbers  can  be  found  approximately  by 
inspection. 

EXAMPLES 

1.  Find  approximately  Vl9. 
19  lies  between  16  and  25. 

Therefore  VT9  lies  between  vT6  and  \/25,  or  between  4  and  5. 
That  is,  \/l9  is  4  plus  a  decimal. 

2.  Find  approximately  V643. 
643  lies  between  400  and  900. 

Therefore,  V643  lies  between  ViOO  and  V900,  or  between  20  and  30. 

That  is,  it  is  20  plus  a  number  less  than  10. 

The  numbers  to  be  added  in  any  case  will  not  change  the  first  figure 
of  the  root  found.  That  is,  by  inspection  we  can  find  exactly  the  first 
figure  of  the  square  root. 

330.  Pointing  off  into  Periods.  Since  102  =  100,  we  know 
that  the  square  root  of  any  number  greater  than  1  but  less  than 
100  is  less  than  10.     Its  integral  part  consists  of  one  figure. 

Since  1002  =  10,000,  we  know  that  the  square  root  of  any 
number  greater  than  100  but  less  than  10,000  is  greater  than 
10  but  less  than  100. 


INVOLUTION  AND  EVOLUTION  245 

That  is,  if  the  given  number  has  3  or  4  digits  in  its  integral  part,  its 
square  root  will  have  2  digits  in  its  integral  part.  If  larger  numbers  are 
given,  the  above  reasoning  can  be  repeated  for  1000^,  etc.,  showing  that 
in  all  cases  if  the  number  be  pointed  off  into  periods  of  2  digits  each  (or 
possibly  fewer  in  the  left  period) ,  then  each  period  will  correspond  to  a 
digit  of  the  root. 

Thus  in  67'62'31,  there  are  three  periods,  hence  there  are  three  places 
in  the  integral  part  of  the  root.  Since  67  lies  between  64  and  81,  the 
square  root  of  67  is  approximately  8,  and  of  676,231  approximately  800. 

ORAL    EXERCISES 

By  the  method  above  state  an  approximate  square  root  of : 

1.  12'36.  3.    l'25'OO.  5.   43'21'00. 

2.  30'95.  4.   8'23'00.  6.   58'61'23. 

331.  When  the  first  digit  of  the  square  root  has  been  found 
by  inspection,  the  process  may  be  continued  thus : 

EXAMPLE 
Find  V2209 : 

1.  The  approximate  value,  as  above,  is  40. 

2.  Let  V2209  =  40  +  7*,  where  r  is  the  rest  of  the  root. 

3.  .-.  2209  =  (40  +  r)2  =  40^  +  2  •  40  •  r  +  r^. 

4.  .-.  2209  -  402  =  2  •  40  •  r  +  r2,  or  609  =  80  ?•  +  r"^. 

5.  Then  609  is  greater  than  80  r,  or  ^f^  =  7  +  dec.  is  greater  than  r. 
.-.  7  +  decimal  is  greater  than  r,  and  it  is  possible  that  1  =r. 
Trying,  we  find  that  80  •  7  +  7^  =  609,  hence,  r  =  7. 

Therefore,  \/2209  =  40  +  7  =  47. 

332.  Thu^,  when  once  an  approximate  value,  a,  has  been 
found  for  the  root,  an  approximate  value  for  the  remainder, 
r,   of    the   root   can   be   found    by   means   of    the    formula: 

1.  Let  n  denote  the  number  whose  square  root  is  sought,  a  denote  the 
approximate  root  at  any  stage,  and  r  the  remainder  of  the  root. 

2.  Then,  w  =  (a  +  r)2  =  a2  ^  2  ar  +  r2. 

3.  .-.  n  -  a2  =  2  ar  +  r'^. 

4.  Or,  n  —  a2  is  greater  than  2  ar,  or,  ^  ~  ^    is  greater  than  r. 

2a 

5.  Hence,      ~  ^    may  be  tried  as  an  approximate  value  of  r. 

2  d 


246 


A  HIGH  SCHOOL   ALGEBRA 


(A) 
Boot 
Number 

18 
182 


9  2 
84'64 

81 

364 
364 


333.    What  precedes  may  be  formulated  into  a  process  or 
working  rule,  thus  : 

1.  Point  off  the  number  into  periods  of  two  figures  each,  he- 
ginning  at  units'  place  {at  the  decimal  point). 

2.  By  inspection  find  the  largest  integer  whose  square  is  not 
greater  than  the  left  period.     (In  Example  A  it  is  9.) 

3.  Use  this  integer  as  the  first  digit  of  the 
root.  Subtract  its  square  from  the  left  period. 
(In  Example  A  this  square  is  81.) 

4.  Bring  down  the  next  period.  (In  Example  A 
this  makes  364.) 

5.  Multiply  the  part  of  the  root  already  found 
by  2.    This  number  is  called  the  trial  divisor. 
(18  in  Example  A.) 

6.  Divide  the  remainder  (omitting  the 
right  digit)  by  the  trial  divisor  and  use  the 
digit  found  as  the  next  digit  of  the  root. 
(In  Example  A,  36  -- 18  =  2.) 

7.  Annex  this  digit  to  the  trial  divisor. 
This  forms  the  complete  divisor.    (182  in  A.) 

8.  Multiply  the  complete  divisor  by  the 
digit  of  the  root  just  found  and  subtract. 

Note.  It  may  happen  that  the  product  to  be 
subtracted  is  larger  than  the  number  from  which  it 
is  to  be  subtracted.  This  indicates  that  the  trial 
divisor  produced  too  large  a  digit.  Try  the  next 
smaller  digit  for  the  figure  of  the  root  last  found. 

9.  Repeat  the  steps  4  ^o  8  until  all  of  the  i^oot 
periods  have  been  brought  down. 

If  the  last  remainder  is  zero,  as  in  Example  B, 
the  process  is  ended,  the  given  number  is  a  perfect 
square,  and  its  root  has  been  found  exactly.  If  the 
last  remainder  is  not  zero,  as  in  C,  the  process 
may  be  continued  as  far  as  desired  by  supplying 
zeros. 


Test.     The  square  of  the  root,  if  complete, 
equals  the  given  number. 


Eoot 

(B) 

3  0.  6  9 

Number 

9'41'.87'61 

6 

9 
41 

60 

4187 

606  3636 

612   55161 

6129  55161 

Eoot 

(C) 
1  4.  1  4+ 

Number 

2'00. 

2 

1 
100 

24 

96 

28 

400 

281 

282 

281 
11900 

2824 

11296 

604 


INVOLUTION   AND  EVOLUTION  247 

334.  When  a  number  contains  a  decimal  the  decimal  point  of 
its  root  is  placed  between  the  figures  furnished  by  the  integral 
periods  and  those  furnished  by  the  decimal  periods  (as  in  C, 
Sec.  333). 


WRITTEN    EXERCISES 

Find  the  square  roots : 

1.   361.       3.   625.       5.   2025.       7.   177,241. 

9.   4,334,724, 

2.   784.       4.   841.       6.    1936.       8.    120,409. 

10.   4,888,521. 

Find  the  square  roots  to  two  decimal  places : 

11.  2.25.  14.   19.36.  17.   2000.  20.  3. 

12.  7.84.  15.   90.25.  18.   0.03.  21.  111. 

13.  6.25.  16.    1.21.  19.    5.  22.  0.00111. 

335.  To  find  the  square  roots  of  fractional  numbers,  either  first 
reduce  the  fraction  to  a  decimal  or  extract  the  square  root  of  both 
numerator  and  denominator. 

WRITTEN    EXERCISES 


Find  the  square  roots : 

1.  ^.       3.  m. 

5-  A%- 

t-  miih 

2-  m-       4.  ^\. 

6-  m- 

8-  «m^f. 

9.  It  is  known  that  if  the  sides  of  a  rectangular  prism  are 
a,  6,  and  c,  the  diagonal,  d,  is  Va^  +  ^^  +  c^.  If  the  sides  of 
such  a  solid  are  20  yd.,  30  yd.,  and  50  yd.,  find  its  diagonal  to 
three  decimal  places. 

10.  Compute  ^s  (s  —  a){s  —  b)  {s  —  c)  to  two  decimal  places, 
when  8  =  18.5  in.,  a  =  10  in.,  6  =  15  in.,  c  =  12  in. 

Solve  and  compute  the  roots  to  three  decimal  places : 

11.  9a;2  =  21.  12.   16.1a;2  =  21. 

336.   Square  Roots  of  Poljrnomials.     The  square  root  of  every 

polynomial  that  is  a  square  may  be  extracted  according  to  the 

process  given  in  Sec.  333.     If  the  polynomial  is  not  a  square, 

the  square  root  may  be  approximated  to  any  number  of  terms. 

17 


248  A   HIGH  SCHOOL  ALGEBRA 

EXAMPLE 

Extract  the  square  root  of  a^  —  2  a5  +  ft^  _  2  ac  +  2  6c  +  c^. 
Boot  ±  (a  —  b  —  c) 

Power 


a2- 

-  2  a&  -  2  ac  +  62  +  2  &c  +  c2 
-2ab             +  &2 

—  2  ac          +  2  &c  +  c2 

-  2  ac          +  2  &c  +  c2 

1.  As  far  as  possible,  arrange  the  terms  according  to  the  descending 
powers  of  some  letter,  as  a  in  this  case. 

2.  The  square  root  of  the  first  term  is  the  first  term  of  the  root.  (Cor- 
responds to  steps  2,  3,  Sec.  333.) 

3.  Divide  the  second  term  of  the  power  by  twice  the  first  term  of  the 
root,  2  a  in  this  case.    The  result  is  the  second  term  of  the  root.  (Steps  5, 6.) 

4.  Subtract  from  the  power  the  square  of  the  binomial  found. 

5.  If  there  is  a  remainder  (as  —  2  ac  +  2  6c  +  c2  in  this  case)  it  shows 
that  the  power  contains  the  square  of  a  trinomial  and  that  there  is  at 
least  another  term  in  the  root. 

This  term  (c)  is  found  hj  dividing  the  remainder  by  twice  the  part  of 
the  root  found  [2(a  —  6)  in  this  case],  for  the  same  reason  as  in  the 
square  root  of  numbers.     (Step  6.) 

6.  The  square  of  the  entire  root  so  far  found  (a  —  6  —  c) 2  must  now 
be  subtracted.  We  have  already  subtracted  the  square  of  the  first  part 
of  the  new  binomial  [(a  —  6)2  in  this  case].  Therefore,  subtract  the  rest 
of  the  square  [—  2(a  —  b)c  4-  c2,  or  —  2  ac  +  2  &c  +  c2]. 

Briefly,  the  trial  divisor,  2(a  —  6),  is  augmented  by  the  next  term, 
—  c,  resulting  in  2(a  —  6)  —  c,  and  this  is  multiplied  by  —  c.  This  gives 
the  part  —  2(a  —  &)c  +  c2,  still  to  be  subtracted.  This  is  analogous  to 
what  is  done  in  extracting  the  square  roots  of  numbers.     (Steps  7,  8.) 

If  there  is  a  new  remainder,  divide  it  by  twice  the  entire  part  of  the 
root  found  and  proceed  as  before.     Test  by  squaring  the  root. 

WRITTEN   EXERCISES 

Extract  the  square  root  of : 

1.  4:9  d'b^-Ua'b  + a*.  4.   a^ -6a^ -\-ll  x^  -  6  x -{-1. 

2.  16xy-{-4:0xyh-\-25yh\        5.   14-4  oj  +  lO  a;2  +  12  a^  +  9  a^. 

3.  4a;4  +  4a^  +  5a;2  +  2a;  +  l.      6.   9  a^  +  U  a^-{-22  x''+12  x-\-9. 

7.  9  «2  +  12  a6  +  4  62  +  6  ac  +  4  6c  +  cl 

8.  1  -  6  a;  +  15  a;2  -  20  a^  +  15  cc^  -  6  a;^  -f  x\ 

9.  x^-4:X^-\-6x^-^2x^-llx'^  +  6x-\-9. 


INVOLUTION   AND   EVOLUTION  249 


10. 

y?      ]p-      z^      xy      yz      xz 

11. 

12. 

4                                   x     x^ 

13.  2  xw  —  4:  xz^ -\- x^  -\- id"^  +  4:  z^  —  ^  iv^. 

14.  a^-{-^b'^-Qdb-Q>bG^2ca  +  c\ 

15.  \a'^-^ah  +  \h'  +  \aG--^hc  +  i-^c\ 

16.  l-122/  +  38/-122/«  +  2/\ 

17.  2xy-2x^y-2yz^  +  x''  +  y^  +  z\ 

18.  a^  +  6a^  +  -2/-a2H-2a4-i. 

19.  xY  —  2  a;^^^;^  4-  4  a^?/^^;  +  9  ^^  —  12  aj.y;^^ 

20.  9  a^  +  25  62  +  9  c2  -  30  a6  + 18  ac- 30  6c. 

21.  m^w^  4-  p2g2  _|_  ^2^2  _|.  2  772,yipg  _|_  2  mn?'s  +  2  pqrs. 

22.  m*  +  5^  4-  my  4-  2  m^^?  -  2  w?q^  -  2  mpq\ 

23.  4  4-  29  a2  _  12  a  -  30  a^  4-  25  a\ 

24.  ^a;^-|.T^  +  ^a;2^4-4a^-2a;?/4-i^2^ 

Extract  the  square  root  to  4  terms  : 

25.  ^2_i.      27.   2/2  _pl.      29.    4  -  «.    *       31.    16  a^  4. 12  a6. 

26.  1  —  x.       28.    a^4-5.       30.    x"'-\-4:y.       32.    9m2  4-9mri. 

REVIEW 
WRITTEN   EXERCISES 
Extract  the  square  root  of : 
1.   14,641.  2.    1.5625.  3.    a'-\-2a'x  +  x\ 

4.  9a44-12a3-20a2-16a4-16. 

5.  a2  4-62  4-4c2-2a64-4ac-46c. 

6.  711"^  4-  4  am^  4-  6  a^^^^s  ^  4  ^3^  _^  ^4^ 

7    i^  _  i  4.  .^ll  _i_  JL  _L  _J_ 
16     42J     20:^2  "^5^3 -^2524* 

8.   a;6  -  6  a;5  4- 15  a:^  -  20  a^  4- 15  a;2  _  6  aj  4- 1. 


250  A   HIGH   SCHOOL   ALGEBRA 

Expand  by  the  binomial  formula : 

9.    {a+,ff.  12.    (a'-\-5y.  15.   (~-^'' 

10.  (ax  +  iy.  13.    (ax2-2  2/y.  .^         ^y 

11.  {ma^-iy.  14.    (Sx'-^y.  ^^'   \2^^~3fj' 

fa      xS}-'^ 

17.  Find  the  middle  term  of  the  expansion  of   f  -  +  -  )  . 

\aj     a) 

18.  Raise  98  to  the  5th  power  by  the  binomial  theorem. 
Suggestion.    Use  100-2  for  98. 

19.  Find  the  ratio  between  the  6th  term  in  the  expansion  of 
[  —^ —  J    and  the  5th  term  in  the  expansion  of  [  — i^ j . 

SUMMARY 

The  following  questions  summarize  the  definitions  and  pro- 
cesses treated  in  this  chapter  : 

1.  What  is  involution  f  Sec.  319. 

2.  What  is  a  binomial  expansion'^  Sec.  321. 

3.  How  else  can  an  expression  in  the  form  of  a  binomial 
expansion  be  stated  ?  Sec.  324. 

4.  State  the  rule  for  writing  a  binomial  expansion.  Sec.  326. 

5.  Define  evolution.  Sec.  327. 

6.  On  what  formula  is  the  general  process  of  finding  square 
root  based  ?  Sec.  332. 

7.  How  may  the  square  root  of  a  fraction  be  found  ? 

Sec.  335. 

HISTORICAL  NOTE 

The  most  important  process  in  involution  is  the  binomial  expansion. 
It  is  the  basis  for  finding  the  powers  of  all  algebraic  polynomials,  and 
likewise  the  basis  of  determining  roots.  The  general  principle  for  writing 
the  terms  of  this  series,  known  as  the  Binomial  Theorem,  was  discovered 
by  Sir  Isaac  Newton,  the  greatest  English  mathematician  of  the  seven- 
teenth century.  Special  cases  of  this  formula,  like  (a  +  &)^  and  {a  4-  &)^, 
were  known  to  the  Hindoos  and  Arabs,  who  used  them  to  find  square 


INVOLUTION  AND  EVOLUTION 


251 


and  cube  roots  of  numbers,  Vieta  in  the  sixteenth  century  knew  the 
expansion  of  (a  +  by,  and  Pascal  constructed  a  table  for  mechanically 
computing  the  numerical  coefficients ;  but  the  full  significance  of  the 
Binomial  Theorem  was  first  discovered  by  Newton  while  investigating  an 
expression  for  the  value  of  tt,  the  ratio  of  the  circumference  to  the  diame- 
ter of  a  circle. 


Newton  was  born  in  a  small  town,  Lincolnshire,  in  1642,  and  his  health 
was  so  delicate  in  his  childhood  that  it  interfered  with  his  education.  It 
is  said  that  his  backwardness 
provoked  the  ridicule  of  his 
companions,  under  the  sting 
of  which  he  soon  surpassed  all 
of  them  in  learning.  He  en- 
tered Cambridge  University  in 
1660  and  readily  mastered  the 
works  of  Descartes,  Vieta,  and 
Wallis.  While  developing 
methods  for  finding  areas 
bounded  by  various  curved 
lines,  he  discovered,  not  only 
the  Binomial  Theorem,  but 
also  the  method  now  known 
as  Calculus,  for  which  Newton 
will  ever  be  famous  as  a  mathe- 
matician. His  discovery  of 
the  law  of  falling  bodies  and 
the  general  principle  of  gravi- 
tation has  placed  his  name 
for  all  time  among  the  fore- 
most men  of  science.  The  English  writer  Pope  has  paid  a  graceful  tribute 
to  Newton's  greatness  in  the  following  couplet  : 

"  Nature  and  nature's  laws  lay  hid  in  night, 
God  said,  '  let  Newton  be '  and  all  was  light." 

But  we  should  not  conclude  from  this  poetic  conception  that  Newton's 
great  achievements  were  mere  flashes  of  thought.  They  were  the  product 
of  a  mind  trained  by  labor  and  informed  by  painstaking  study.  Genius 
pointed  out  the  way  to  greater  things,  but  not  till  Newton  had  climbed 
the  height  of  other  men's  knowledge. 


Sir  Isaac  Newton 


CHAPTER   XXI 

RADICALS   AND   EXPONENTS 

DEFINITIONS   AND    PROPERTIES 

337.  Rational  Numbers.  Integers  and  other  numbers  expres- 
sible as  the  quotient  of  two  integers  are  called  rational  numbers. 

Thus,  5  and  |  are  rational  numbers. 

Also,  .2,  which  is  expressible  as  -^-^^  is  a  rational  number. 

338.  Irrational  Numbers.  Any  number  not  rational  is  called 
an  irrational  number. 

Thus,  \/2,  VS,  v/10,  —-i ,  1  4-  V3,  V2  —  VS,  are  irrational  numbers. 
v5 

Numbers  like  tt  =  3.14159+,  the  ratio  of  the  circumference  to  the 
diameter  of  a  circle,  are  also  called  irrationals. 

A  special  class  of  irrational  numbers,  namely,  even  roots  of  negative 
numbers  are  called  imaginaries.  We  have  seen  that  (  +  2)  (  +  2)  =4,  and 
that  (-  2)  (-  2)  =  4  ;  hence,  V4:=  ±2,  similarly  V3  =  ±  1.732+  ;  but 
we  have  not  yet  found  two  equal  numbers  whose  product  is  —  4,  or  any 
other  negative  number.  Hence,  the  use  of  numbers  like  V—  4  and  V—  a, 
will  not  appear  in  the  processes  or  in  the  roots  of  equations  until  Chapter 
XXVIII  has  been  studied. 

339.  An  indicated  root  of  any  number  is  called  a  radical. 

Thus,V5,  v^8,  \/  — ,  Va  +  x^,  are  radicals. 
'  3  & 

In  the  present  chapter  all  roots  that  cannot  be  exactly  extracted  by 
inspection  are  indicated. 

340.  Although  all  indicated  even  roots  may  be  taken  either 
H-  or  — ,  it  is  customary  in  the  treatment  of  radicals  to  omit 
these  signs,  regarding  the  radical  as  positive. 

252 


RADICALS   AND   EXPONENTS  253 

341.  Surd.  A  monomial  containing  an  indicated  root  of  a 
rational  number  is  sometimes  called  a-  surd.  The  part  under 
the  radical  is  called  the  radicand. 

342.  The  number  denoting  the  root  to  be  taken  is  called  the 
index  of  the  root. 


Thus,.  \/2,  v5,  3  \/7,  5  Vl4  —  a,  are  surds  the  radicands  in  order  are 
2,  5,  7,  14  —  a,  and  the  indices  of  the  rdots  are  2,  3,  5,  2. 

343.  A  radical  expression  with  no  rational  factor  is  called 
an  entire  surd ;  otherwise  a  mixed  surd. 

344.  The  index  of  the  root  is  called  the  order  of  a  surd. 

For  example,  \/2  is  of  the  second  order,  V2  is  of  the  third  order,  v^3 
is  of  the  fifth  order. 

345.  A  surd  of  the  second  order  is  also  called  a  quadratic 
surd,  of  the  third  order  a  cubic  surd,  and  one  of  the  fourth 
order  a  biquadratic  surd. 

346.  An  expression  involving  one  or  more  radicals  is  called 
a  radical  expression. 

Thus  5  4-  2  V3 1»      =r —  are  radical  expressions. 

'  '       Vx  2-V36 

347.  Some  Properties  of  Radicals.  A  few  important  proper- 
ties of  radicals  are  given  here.  The  fuller  treatment  is  con- 
tained in  Chapter  XXVI  on  Exponents. 

348.  I.    Va.  V6  =  V^. 

For  example,  V2  •  V3  =  a/6. 

That  this  is  true  may  be  seen  by  squaring  both  members. 

Thus,  (V2  .  \/3)(V2  .  V3)  =  \/6  .  V6, 

or,  \/2  .  \/2  .  \/3  •  V3  =  V6  .  V6, 

or,  2-3  =  6,  which  is  known  to  be  true. 

In  the  same  way,  it  may  be  seen  that  for  every  a  and  b,  Va  •  Vb  =  Vab. 

In  words : 
K   The  product  of  two  square  roots  is  the  square  root  of  the  product 
of  the  numbers. 


254  A  HIGH  SCHOOL  ALGEBRA 

WRITTEN  EXERCISES 

Show  by  squaring  that : 

1.  V3.V5  =  V15.  5.    V2a.  V36=V6a&. 

2.  V4.V7=V28  6.    V^.V5^  =  V5^. 

3.  V3.V7=V2r.  7.    V2.  V3.  \/5  =  V30. 

4.  V5  •  Vil  =  ^55.  8.    Va  •  V&  •  Vc  =  Va6c. 

349.   IL    ^a^  =  Va^  Vb  =  aVo. 
In  words : 

Factors  which  we  perfect  squares  may  he  taken  from  under  the 
radical  sign. 

Thus,  Vl8  =  V9  •  V2  =  v^  .  \/2  =  3\/2. 

WRITTEN    EXERCISES 
Take  all  factors  which  are  perfect  squares  from  under  the 
radical  sign : 


1.  V20. 

2.  V27. 

5. 
6. 

7. 
8. 

V45. 
V75. 
V24. 
V32. 

V6  = 

9.    VI2. 

10.  V40. 

11.  V500. 

12.  Vl28. 

13. 
14. 
15. 
16. 

V8ci2. 

3.  V50. 

4.  V48. 

350.   III. 

V45ay. 

In  words : 

Any  factor  outside  the  radical  sign  may  he  placed  under  the 
radical  sign  provided  the  factor  is  squared. 

Thus,  3^/2  =  Vg  •  \/2  =  Vl8. 

WRITTEN    EXERCISES 

Place  under  one  radical  sign  : 

1.  6V2.        5.   3  .  V7  .  2.  9.   4V2  •  3.     13.    t^'g. 

2.  5V3.        6.    5  .  V3  .  V2.       10.    6V2.  14.   r^Vr. 


3.   2.V3.     7.    2.V3.V11.     11.   2x^3x.     15.    ^VlSx?/. 


4.   5  •  V'7.     8.    5  •  V3  .  V7.       12.   ah^hc.       16.    aV&  —  a. 


RADICALS  AND  EXPONENTS  255 

351.  Fractional  Exponents.  A  more  convenient  notation  to 
indicate  roots  is  found  in  fractional  exponents.  In  order  to  find 
the  proper  meaning  of  such  exponents,  we  shall  assume  that  the 
laws  of  integral  exponents  apply  also  to  fractional  exponents. 

Assume  the  law  «"*•«"*  =  «"»+"  to  hold  when  m  =  \  and  n  =  \, 
Then,  ah  .  a^  =  a^^^-  =  a^  =  a. 

Therefore,  a^  is  one  of  the  two  equal  factors  of  a,  or  the  square  root 
of  a. 

(1)  Hence  we  may  write  Va  =  a^. 
Similarly,  a^  -  a^  -  a^  =  a^+i+i  =  «!  =  a. 
Hence,  a^  is  the  cube  root  of  a. 

(2)  and  v^  =  a^. 

352.  In  general  Va  =  a". 

EXAMPLES 

1.  9^  =  V'9=3.  3.    27^=\/27=3. 

2.  (4a2)^=V4^  =  2a.  4.    (8  a^b^)^  =  y/Scfib^  =  2  ab^. 

WRITTEN    EXERCISES  ,  -^ 

j 

Write  with  radical  sign  and  simplify  : 

1.  4i      3.    (9a;2)J.      5.    (4:y'')K        7.    Si  9.    (a^f)^- 

2.  #.     4.    {xY)K      6.    (25«2)i      8.    {27a')K      10.    (27  66)i 

353.  Applying  the  law  of  multiplication  to  a^  •  a^. 
we  have  a^'^^  =  a^. 

But,  a3  .  a^  =  (a^)2. 

Therefore,  J  =  (afy  =  (\/a)2. 

Similarly,  a^  -  a^  -  a^  —  (a^y  =  ( Va)^ 

354.  In  general  the  numerator  of  the  fractional  exponent  de- 
notes the  power  to  be  taken,  and  the  denominator  denotes  the  root. 

m  

In  symbols,  a«  =  (Va)"*. 


256  A  HIGH   SCHOOL   ALGEBRA 

EXAMPLES 

1.  8^=(\/8)2  =  22  =  4. 

2.  165=(\/16)8  =  23  =  8. 

3.  9^=(\/9)3  =  27. 

4.  (4  a264)^  =  (2  a&2)5  ^  32  ^sftio. 


WRITTEN    EXERCISES 

Write  with  radical  sign : 

1.   al 

5.    (2a)i 

9. 

ahi. 

2.   bi 

6.    (0^2/)^. 

10. 

5  xi(y  +  z)i. 

3.    A 

7.    Sbi 

11. 

2rhi. 

4.    x\ 

8.   5ma;i 

12. 

1    m 

5a'b\ 

FiD 

id  the  rational  values  of : 

13. 

uK 

20.    32I 

27.    (f)l 

14. 

27i 

21.    (-27)1 

28.    9^(i)i 

15. 

leK 

22.    (-125)1 

29.    16^(i)i 

16. 

27t. 

23.    (i)\ 

30.    27^(4)^: 

17. 

9i 

24.    (i)i 

31.    ^64.  (,V)^. 

18. 

si 

25.    (1)1 

32.   (-125)VS. 

19. 

64l 

26.    (-/^)i 

33.   (-343)IVS- 

Wr 

ite  with  fractional  exponents  and 

simplify  when  possible : 

34. 

V^. 

40.   3^/4  m2. 

46.    5a2^3a3. 

35. 

■V2x. 
S^/x, 

41.  5^8  2^. 

42.  2^21  xf. 

47.   xViyzy, 

36. 

48.   abV{a-j-by, 

37. 

V^. 

43.   3A/a263. 

49.    2v5^ .  ^/g^: 

38. 

V4a. 

44.    ^-^ab\ 

50.    V^  .  V6^. 

39. 

5V25  62. 

45.    -2a^32a3, 

51.    S/a^  •  Va^. 

RADICALS   AND  EXPONENTS  257 

PROCESSES 

355.  Reduction.  The  reduction  of  radicals  means  the  chang- 
ing from  one  form  to  another  equivalent  form,  not  necessarily 
to  a  simpler  one. 

1.   To  reduce  a  mixed  surd  to  an  entire  surd. 

EXAMPLES 

1.  3V5=V32T5=v'i5.  Sec.  354. 

2.  2v^  =  ^23  .  3  a  =  \/24^. 

To  reduce  a  mixed  surd  to  an  entire  surd  raise  the  rational 
factor  to  a  power  equal  to  the  index  of  the  root  and  place  the 
result  under  the  radical  sigyi  as  a  factor. 

WRITTEN    EXERCISES 

Reduce  to  an  entire  surd : 

1.    -1V2.  9_    ^J[^  16.    {a  +  h)^ab.    ■ 


1 


5\a 


^-    a^^-  10.  x-^^y.  17.   2(a  +  h)^^~^. 

3.  |V^.  11.  2^3  ""^^ 
^     _                                1      

4.  2^2.  12.  i^3|.  ^^-    ^Va-6. 

5.  3^5.  13.  i^2f. 


6.    a^a.  14.    |V6^.  '    «-^^'a+6 


7.    aVa6.  3/1  a;  4- 2    / 

15.    a&^-^-  20.    -^-kJI-- 

\abc  x-Z\        X 


8.    2V-y-a;2.  \a6c  a;-J\        a; +  2 

2.   To  change  the  order  of  a  surd. 

EXAMPLES 

1.  VS=\/W^  =  </9. 

2.  S^2^^  =  S\/¥^=sV8cfi. 

To  multiply  or  divide  the  order  of  a  surd  by  a  positive  integer 
multiply  or  divide  the  exponent  of  each  factor  under  the  radical 
iign  by  the  same  number. 


258  A  HIGH   SCHOOL   ALGEBRA 

This  process  is  made  easier  by  using  fractional  exponents. 
Thus,  to  change  -V2a^  to  the  ninth  order, 

WRITTEN    EXERCISES 

1.  Eeduce  the  radicals  to  the  fourth  order  : 

V2,    Va,    2Vab,    Vl,    2^2^'    « Va  +  6. 

2.  Eeduce  the  above  radicals  to  the  sixth  order. 

3.  Eeduce  the  radicals  to  the  ninth  order : 

4.  Eeduce  the  radicals  to  the  second  order : 

3.   To  reduce  radicals  to  the  same  order. 

Apply  the  processes  in  case  2  above  to  reduce  each  radical  to  the 
given  order. 

WRITTEN   EXERCISES 

1.  Eeduce  to  radicals  of  the  eighth  order :  V3,  V2. 

2.  Eeduce  to  radicals  of  the  sixth  order :  VS,  V4. 

Eeduce  to  entire  surds  of  the  order  named : 

3.  V2,    v^3  a,  8th  order.  6.    V|,    'V\x^y,  8th  order. 

4.  V5     2Va,  12th  order.         7.  p^ph',  ^^ax,  12th  order. 

5.  3-C/aV,  Vmn,  10th  order.     8.    6  VS,   .3a/.5",  6th  order. 


9.   Eeduce    V\  a'^b,   Ta/^—-^,  to  the  15th  order. 
10.   Eeduce    V2,    ^3,    ^4,    ^^5,  to  the  20th  order. 


RADICALS   AND  EXPONENTS  259 

11.  Eeduce  to  radicals  of  the  sixth  order : 

V2,    -y/ab,    Va  +  b. 

12.  Reduce  to  radicals  of  the  tenth  order : 

V2,    -Vaxj  5-V2xy,    -^a  +  b,    {a—b)-\/x  —  2y. 

Reduce  to  radicals  of  the  same  order : 

13.  {x-{-y)^x  —  y,  -\/x  —  y.        16.    ^/a,    ^/ab,    ^a?W. 

14.  Va,    -y/o}  -  b%    y/a-b.       17.   Vx,    ^xy,    Vx  -  y. 

15.  V^,  v'^iT^,  ^7^^^^.         18.    V^^^^,  Va-6  +  c. 
Which  is  the  greater 

19.    -v^orv^S?         20.    V2or-v/3?         21.    V2|  or  ^4  ? 

4.  To  reduce  a  radical  to  its  simplest  form. 

EXAMPLES 
1.    V8a3  =  ^4  a2  •  2  a  =  2  a  V2a.  Sec.  349. 


2.  V2a6x5"=  V(a-^x)3  .  2 a;^  =  a^ V2 a;^. 

3.  V|=V|  =\/fT2  =  |V2. 

4.  \/^^=  ""^(0X2)^  =  v/^. 

Note  that  different  exponents  like  the  2  and  3  in  Vcfib^  must  not  be 
canceled  from  the  common  index. 

356.    The  reduction  of  radicals  is  facilitated  by  the  use  of 
fractional  exponents. 
Consider  Example  2  above : 

V¥ch^  =  (2  a^x^)^  =  2^(a6)^  •  (x^)^  =  2^  a^  •  (0^  -  x^)^ 
=  23  a^x (x2)i  =  a^x{2  x^yl  =  a^xy/2c^. 

Similarly,  for  Example  3  : 

v/T=(^)^=(i.2)^  =  K2)^=^V2. 
Also  for  example  4  : 


260  A   HIGH  SCHOOL  ALGEBRA 

357.    A  radical  is  understood  to  be  in  its  simplest  form  when  : 

(1)  No  factor  can  he  taken  from  under  the  radical  sign  {Ex- 
amples 1  and  2,  Sec.  355,  4) ; 

(2)  The  radicand  is  integral  {Example  3). 

(3)  The  radicand  is  not  a  power  whose  exponent  has  a  common 
factor  with  the  order  of  the  radical  {Example  Ji). 

WRITTEN  EXERCISES 

Reduce  to  simplest  form  : 

11.   -v/320. 


1. 

V18. 

2. 

V27. 

3. 

V20. 

4. 

V32. 

5. 

2V8. 

6. 

3V12. 

7. 

5V50. 

8. 

-^16. 

9. 

3\/32. 

10. 

3V75. 

26. 

a     W- 
b-'Ma' 

12. 

a/3000. 

13. 

Vi 

14. 

3Vi. 

15. 

3V?;. 

16. 

2</l 

17. 

5</T,. 

18. 

^a?x^y. 

19. 

yl'9- 

20. 

4- 

21. 

V£- 

22. 

'ih- 

23. 

4A/15i 

24. 

ws- 

25. 

'-^a'^b^. 

"81  35. 


34.    V184-9V2. 

^         2 

a^  +  52 


ab 


28.  Vl-(i)'.  36.   abyl2-{- 

29.  vi  +  g?-  37.  ^^^^3^. 

30.  V3  -  ( j)^  38_    VT^'  -Uxy  +  7  y\ 

31.  Vl-(l)^  ^^     {^x'-lOax  +  ^a'f. 

32.  3J— .  40.    ^m'-Q{m^-Qm^)^. 


27 
33 


;.  ^p 


41.    Va -^  6  .  Va^c  H- 62. 


42.    (a-6)Va2-2a6  +  62. 


43. 

^i-H!)' 

44. 

x'.-(f)* 

45. 

3/    10 

RADICALS   AND   EXPONENTS  261 


47.  V(a^  -  2a%  +  a'b')  -^  h\ 

48.  a6V3a;2-|-6a^2/  +  3  2/2. 

49.  {x-\-y)-\/o?h-a\ 


50.   V.+(fJ- 


46.    ^27-8lV3.  "•  \\^  +  ^A'-i 

358.  Addition  and  Subtraction  of  Radical  Expressions.  Radi- 
cals can  be  united  by  addition  or  subtraction  only  when  the 
same  root  is  indicated  and  the  expressions  under  the  radical 
sign  are  the  same  in  each. 

359.  When  the  expression  cannot  be  put  into  this  form  the 
sum  or  the  difference  can  only  be  indicated. 

360.  To  add  or  subtract  radical  expressions  having  the  same 
radical  part,  add  or  subtract  the  coefficients  of  their  radical  xx^i'ts. 

For  example : 

1.   2  V3  +  3  \/3  =  5  V3.      2.2  VU  +  VSOO  =  4  VS  +  10  V3  =  14  V3. 

3.    Add  \/2,  -V8,  ^16,  v^^^54: 

_V8=-2\/2;   ^/m  =  2^2',   </^^^^  =  - 3\/2. 
.-.  V2  -  \/8  +  ^/l6  +  ^/T^  =  V2  -  2  \/2  +  2  v^2  -  3  v^2  =  -  V2  -  v^. 

Since  only  like  radical  parts  may  be  added,  it  will  be  clear  that  Va  +  Vb 
does  not  mean  Va  +  b.    Show  this  by  letting  a  =  9,  6  =  16. 

WRITTEN    EXERCISES 
Find  the  sum : 

1.  V2,  V8,  V18.  8.  V6,  V24,  V63. 

2.  V75,  -Vl2,  -V3.  9.  Vl08,  -  Vi2,  V48. 

3.  V8,  V5,  -  Vl8.     '  10.  V75,  Vis,  -  V27. 

4.  V128,  -2V50,  V72.  11.  V80,  V20,  -  V45. 

5.  ViO,  -V320,  Vl35.  12.  V44,  -  V99,  Vi2l. 

6.  8  Vis,  -iVi2,  4V27.  13.  5V2i,  -  VSi,  3  V96. 

7.  V72,     -3V9,    6V2i3.  14  \/27^,  -  V6i¥^  Vl6^. 


262         A  HIGH  SCHOOL  ALGEBRA 

361.  Multiplication  of  Radical  Expressions  containing  Square 
Roots.  In  multiplying  expressions  containing  indicated  square 
roots,  make  use  of  the  relation  va  •  Vfe  =  Va5. 


1. 


EXAMPLES 

3- 

4V5 

2. 

2- 

-     V3 

6  + 

2\/3 

5  +  2\/3 

18- 

24  V' 5 

10- 

-5V3 

6V3- 

-8\/l5 

4\/3- 

-6 

18- 

24V5  + 

6V3- 

-8\/i5. 

10- 

-     V3- 

-6 

=  4  -  \/3. 


WRITTEN    EXERCISES 

Multiply : 

1.  2-hV5  by  2--V5.  7.  4-hV5  by  VlO. 

2.  1  + V3  by  2-fV5.  8.  3-Vl5  by  2  +  V5. 

3.  2-h  V3  by  2-h  V3.  9.  l-hV2  by  1-V8. 

4.  V2-f  V3  by  1- Va.  10.  2V3-3V5by  V3-V5. 

5.  V3-  V5  by  V3  + VK  11.  Vi4  +  V7  by  V8-V21. 

6.  V5-^6  by  ^-V6.  12.  V5-V48  by  V5  +  V12. 

362.   Division  of  Square  Roots.     The  quotient  of  the  square 
roots  of  two  numbers  is  the  square  root  of  the  quotient  of  the 


numbers.     In  symbols,  — ^  =  a/^. 


V6 

Thus,  -^  zza/-,  because,  multiplying  each  member  by  itself, 
^M       '  6 


or 


V5     \/5_^,'5     ^/5 


\/5  .  V5 
V6  .  V6 


Vi•^| 


s=v(ir. 


or  -  =  - ,  which  is  evidently  true. 

6      Q' 


RADICALS   AND   EXPONENTS  263 

ORAL   EXERCISES 

Read  each  of  the  following  as  a  fraction  under  one  radical 
sign : 

1.    ^.  3.    ^.  5.   ^.  7.     ^^ 


V5  V7  .  V5  Vl5 

2.    ^.  4.    J-  =  ^.  6.    ^.  8.    ^. 

V7  V2      V2  V8  V20 

363.  In  the  four  processes  with  radicals  explained  in  Sees. 
359-362  we  have  admitted  only  the  exact  values  of  the  radical 
expressions,  but  if  we  accept  approximate  values  of  the  radi- 
cals, we  may  further  simplify  the  results  obtained,  and  this 
is  often  done  for  practical  purposes. 

EXAMPLE 

Each  of  the  two  legs  of  a  right  triangle  is  1  in.,  and  the 
hypotenuse  is  V2  in.     Find  the  perimeter. 

The  exact  result  is  (1  +  1  +  V2)  in.  =  (2  +  \/2)  in. 
If  we  accept  an  approximate  result,  taking  three  decimal  places  as  the 
degree  of  accuracy,  we  have 

(2  +  1.414)  in.  =  3.414  in.  Sees.  333,  334. 

WRITTEN    EXERCISES 

Find  to  two  decimal  places  the  value  of : 

1.  2+V3.  4.   4-V2.  7.   iV2-l. 

2.  3-V3.  6.    V2+V3.  8.   iV3-f2. 

3.  3-V2.  6.    V3-V2.  9.    V5-2. 
Find  to  three  decimal  places  the  value  of : 

*10.    iV3.  12.   ^~^.  14.    V5-V2. 

3  2 

11.   |V3-iV2.        13.    V2-V3.  15.    V6-V3 

16.  Find  x  to  two  decimal  places  in  the  equation  a;^  —  3  =«  0. 

17.  The  altitude  of  an  equilateral  triangle  with  sides  1  in.  is 
i  \/3   in.     Find  the  altitude  to  two  decimal  places. 

18 


264  A  HIGH   SCHOOL   ALGEBRA 

18.  Find  the  perimeter  of  a  right  triangle  to  two  decimal 
places  whose  sides  are  1  in.,  2  in.,  and  V5  in. 

19.  Find  the  perimeter  of  a  right  triangle  to  two  decimal 
places  whose  sides  are  1  ft.,  ^  ft.,  and  ^  V3  ft. 

364.  Rationalizing  the  Denominator.  Multiplying  both 
numerator  and  denominator  of  a  fraction  by  an  expression 
that  will  make  the  denominator  rational  is  called  rationalizing 
the  denominator. 

Thus,  multiplying  both  numerator  and  denominator  of  ^^  ^y  V2,  we 
obtain  V2 

V5  ^  \/2  •  \/3  ^  \/6  ^ 

V2  ~  V2  .  V2        2    ' 

•    WRITTEN    EXERCISES 
Rationalize  the  denominator  of : 

1. 
2. 


1 

4.   2. 

7.  ^. 

10.  1« 

V2 

V3 

V7 

V5 

1 

3 

5 

,,   6 

5.    ■ 

8.     • 

11.  — 

V3 

V7 

V3 

V3 

1 

6.  ^. 

9.  ^~'. 

12.  « 

VB 

V3 

V3 

V2 

365.  Rationalizing  Factors.  When  the  denominator  is  of 
the  form  Va  +  V6  or  a  4-  V&,  the  rationalizing  factor  is  the 
same  binomial  with  the  connecting  sign  changed,  often  called 
the  conjugate  binomial. 

It  is  not  necessary  in  elementary  algebra  to  take  up  the  rationalizing 
of  more  complicated  denominators. 

EXAMPLES 
1.    Rationalize  the  denominator  in 


2-V5 
The  conjugate  of  2  —  VS  is  2  +  Vb. 

Then,  ^-  =        S(^+^^^ =  6  +  «VS=-(6  +  3V5). 

2-v'5      (2-V6)(2  +  V6)         4-6 


RADICALS   AND  EXPONENTS 


265 


2.    Eationalize  the  denominator  in 

The  conjugate  of  VS  +  VE  is  VS  —  Vo. 
Then      2+V3    ^    (2  +  V3)(  V3  -  V5) 


V3 


V3+V5 


V3+\/5 


(V3+V6)(\/3-V5) 
3  4-  2  V3  -  2  VS  -  VT5 


2  V3  +  3-  2V5-\/l5 
3-5 


3.    Rationalize  the  denominator  of 


3-V5 


V2-V5  +  V3 
First  multiply  the  terms  by  the  expression  V2  +  (  VS  —  \/3). 
Then,  we  have  (3  -  v^5)(  V2  +  V5- V3)  ^  (3  -  V5)  (  v^  +  V5- V3) 

2-(\/5-\/3)2    ^  -6  +  2V15 

Then,  multiply  by  the  conjugate  —6  —  2  \/l5  and  simplify  as  usual. 


WRITTEN    EXERCISES 
Rationalize  the  denominators : 


O 


ir  ->" 


4l 


f 


2  4-V3 
3+V3 

3V3+2V2 
V3-V2 
1 


4. 


1-V2 
3 


X 


14. 


15. 


16. 


V5 
1 


5. 


6. 


7. 


8. 


V3  +  V7 
2-V5 


V5 
3 


V5+V2 
V5  +  2  V2 


1-V2+V3 

1+V2 
V3-V5  +  4 

V2-V3 
V2-V3+V5 
3-V5+V7, 
3  4-V5-V7 


2V2 


17. 


9. 


10. 


11. 


12. 


3+V5 
3-V5* 

8-5V2^ 
3-2V2' 

2  +  4V7^ 
2V7-1* 
2V15-6 

V5  +  2  V2 


18. 


19. 


20. 


4-(V6  +  V5) 
c 


-\a  —  h 
1 


a  —  V2  6  +  c 

2  Va  —  V6  -f  Vc 


266  A  HIGH   SCHOOL   ALGEBRA 


2^     Va^-l+Va^  +  l.  23     ^x" -j-x -\-l -1 


22    «-V«'-i-  24.  Vi^  +  g-Vi>-g. 

a  +  Va^  — 1  "  Vp  +  g+Vp  — g 

366.  Chapter  XII  dealt  only  with  rational  factors,  but  the 
properties  of  radicals  make  it  possible  to  find  irrational  factors. 

EXAMPLES 

1.  Factor:  2a;2_l. 

This  is  the  diflference  of  two  squares,  if  we  admit  radicals,  because  2  is  the  square 

ofN^. 
Then,  2a;2- l=(\/2cK-l)(\/2a;+ 1). 

2.  Factor:  x^  —  5. 

Using  radicals,  X^  —  5  =  X^  —  (\/5)2, 

Then,  a:2 -  5  =  (X  -  V5) (x  +  V6). 

3.  Similarly,  x  —  y=  (Vcc  +  -Vy){Vx —-Vy). 

4.  x—y  —  {Vx  —  ^y){-Vx^  +  Vxy  +  -^y^). 

367.  Irrational  factors  may  be  used  to  solve  equations. 
Thus,        In2a;2_i=o,  (V2x  -  l)(V2x +  1)  =  0. 

Therefore,      V2x  —  1  =  0,   and  V2x  +  1=0. 
Solving, 


cc=  — =  -V2,  also  x=- -^=-1^/2, 

V2      2       '  V2  ^ 


WRITTEN    EXERCISES 
Factor : 

1.  x'^-S.  3.   8a;2_l.  5.    6x''-9.  7.   z'^-7. 

2.  4aj2-5.         4.   5a;2_4,  g.    2a^-b.  8.    a-^^. 

9.    What  factor  taken  with  ^/a  —  -y/b  makes  a—  b? 

10.  What  factor  taken  with  a;  —  ^2  makes  a^  —  2  ? 

Solve. 

11.  2/2-2=0.  13.    2j92_l=0.  15.   2m2-5  =  0. 

12.  ;22_3  =  o.  14.    ax^-b  =  0.  16.    2a;2-12  =  0. 


RADICALS   AND   EXPONENTS  267 


REVIEW 

WRITTEN   EXERCISES 

Simplify : 

1.     V5 

4      VlO. 

^     3V3  4-2V2 

V60 

V40 

V3-  V2 

2.    VV- 

-Vf 

5.    V50  +  V128. 

8.    V6-V2. 

3.    V6. 

\/l25. 

6.    V3^V5. 

9.    (2  +  V3)2. 

10.   (5+V7)(5- 

-V7).                  11.    (2V3 

-f-3V5)--Vl5. 

12.    (V6+Vl5)(V8-V20). 

Express  with  rational  denominators,  and  with  at  most  one 
radical  sign  in  the  dividend : 
13.    Vl2^V3. 

14.  VT^vn. 

15.  2V24-r-2V6. 

16.  2--3V5. 

4 

17.  —^ — . 
V5-1 

18.  1  --(V2-10). 

19.  V2-(V2-V3). 

20.  (2V6  +  5VT2)--V6. 

21.  (5Vi8-8V50)--2V2:         ~^'    5V2  +  2V7* 

Solve  by  factoring : 

27.  3a^-l  =  0.  29.    5  a;^  -  a  =  0. 

28.  4^2-2  =  0.  30.    3a;^-9a;  =  0. 

31.  Computing  the  square  root  to  two  decimal  places,  find  r 
in  the  equation  irir  =  8.    (Use  tt   =  3.1416.) 

32.  Find  t  to  three  decimal  places,  using  g  =  S2,  in  the 
equation  48  =  i^il 

33.  Find  the  value  of  ^  7i  (J5  +  6  +  V^),  taking  7i  =  5,  6  =  3, 
^  =  8,  and  computing  the  radical  to  two  decimal  places. 

34.  Find  the  value  of  the  expression  in  Exercise  33  when 
^  =  9,  b  =  S,  B  =  17,  computing  the  radical  to  3  decimal  places. 


22 

8-5V2 

3-2V2 

23. 

3  4-V5 
3-V5 

94. 

V3 

V5-V3 

2.*i 

2  4-4V7 

2V7-1 

Oft 

4V7  +  3V2 

268  A  HIGH  SCHOOL  ALGEBRA 

SUMMARY 

The  following  questions  summarize  the  definitions  and  pro- 
cesses treated  in  this  chapter : 

1.  Define  ratioyial  numbers;  also  irrational  numbers. 

Sees.  337,  338. 

2.  Define  a  radical;  also  a  surd;  also  a  radical  expression. 

Sees.  339,  341,  346. 

3.  What  sign  is  chosen  for  radicals  of  even  order  ? 

Sec.  340. 

4.  What  is  the  radicaiiff .?     Illustrate.  Sec.  341. 

5.  What  is  an  eyitire  surd?     A  mixed  surd?  Sec.  343. 

6.  What  is  meant  by  the  order  of  a  surd  or  radical  ?    State 
a  quadratic  surd ;  a  cubic  surd ;  a  biquadratic  surd. 

Sec.  344,  345. 

7.  What  is  another  expression  for  the  product  of  two  square 
roots  ?  Sec.  348. 

8.  How  may  factors  be  placed  under  the  radical  sign  ? 

Sec.  350. 

9.  How  is  a  mixed  radical  reduced  to  an  entire  radical  ? 

Sec.  355. 

10.  Illustrate  how  to  change  the  order  of  a  radical. 

Sec.  355. 

11.  State  how  to  simplify  a  radical.  Sec.  355. 

12.  Illustrate  how  the  use  of  fractional  exponents  simplifies 
the  process  of  reduction  of  radicals.  Sec.  356. 

13.  What  radicals  may  be  added?     How  are  they  added  or 
subtracted  ?  Sees.  358-360. 

14.  What  is  another  expression  for  the  quotient  of  two 
square  roots  ?  Sec.  362. 

15.  What  is  meant  by  rationalizing  the  denominator  of  a  frac- 
tion containing  radicals  ?  Sec.  364. 


CHAPTER  XXII 

QUADRATIC  EQUATIONS 

/ 
RATIONAL 

368.  The  general  form  of  a  quadratic  equation  is 

(1)  ax^ -{- bx -\- c  =  0. 

By  dividing  this  equation  by  a,  the   coefficient  of  x"^,  the 

b        c 
equation  becomes  a;^  +  -a;  +  -  =  0. 
a        a 

b  c 

Putting  p  =  -,  g  =  -  to  replace  the   fractional   forms,  the 
a  a 

equation  becomes 

(2)  x'^+px-\-q=0 

which  is  also  a  general  form  for  quadratic  equations. 

369.  Kinds  of  Quadratic  Equations.  The  equation  ax^  -\-  bx 
+  c  =  0  is  said  to  be  a  complete  quadratic  equation  when 
neither  b  nor  c  is  zero ;  that  is,  when  there  are  three  terms, 
one  containing  ic^,  another  x,  and  a  term  without  x  (absolute 
term).  In  any  other  case  it  is  called  an  incomplete  quadratic 
equation. 

370.  A  quadratic  in  which  the  first  power  of  the  unknown 
does  not  occur  is  called  a  pure  quadratic,  and  one  in  which  it 
occurs  is  called  an  afifected  quadratic. 

Thus,  a:2  +  5a;  —  2=:0isa  complete  quadratic  equation,  while  2x^  —  5 
=  0,  and  ic2  +  7  X  =  0  are  incomplete  quadratics.  The  second  equation  is 
a  pure  quadratic  and  the  others  are  affected  quadratics. 

371.  Solution  by  Completing  the  Square.  The  following  ex- 
ample shows  a  general  method  for  solving  quadratic  equations 
with  one  unknown : 

269 


270  A  HIGH   SCHOOL  ALGEBRA 

EXAMPLE 

Solve:  a;2-f8x  +  7  =  0.  (1) 

Transposing  the  absolute  term,  X'^  -{-8  X  =  —  7.  (2) 

Making  the  left  member  a  square     ic^  4-  8  X  +  16  =  9  (3') 

by  adding  16  to  both  members,  ^         ^  •  \^   j 

.'.(x  +  4y  =  9.  {4} 

Extracting  the  square  root  of  both  ^    \   a        \    o  /  e\ 

members,  X  +  ^  -  ±  6.  {5) 

...  a;  =  -l  or-7.  (6) 

The  process  consists  of  two  essential  parts : 

(1)  Making  the  left  member  a  square  while  the  right  member 
does  not  contain  the  unknown. 

This  is  called  completing  the  square. 

It  is  based  upon  the  relation  (x  +  a)^  =  x^  +  2  ax+  a^  (Sec.  141,  p.  103) 
in  which  it  appears  that  the  last  term,  a^,  is  the  square  of  one  half  of  the 
coefficient  of  x. 

(2)  Extracting  the  square  roots  of  both  members  and  solving 
the  resulting  linear  equations. 

WRITTEN   EXERCISES 

Solve  completely: 

1.  a;2  +  8a;  =  9.  %.    x'^-Q>x=l, 

2.  X^+4.X  =  12.  9.    22  _  40  2;  =  41. 

3.  a)2_|_i2a;  =  -ll.  10.    w^-w  =  %. 
A        2       c           a                                    11.    ^2_  2^  =  8. 

12.    u^-  u  =  -  1 

5.  a;2+i0a;  =  ll.  ^ 

6.  x^  +  5x  =  -^^.  13-   v^  +  3v=^. 

7.  i»2_20a;  =  -75.  14.   s2-18s  =  19. 

372.  Approximate  Roots  Expressed  Decimally.  Square  roots 
which  cannot  be  found  exactly  may  be  indicated  and  should 
be  so  used  for  checking,  but  the  roots  of  quadratic  equations 
expressing  the  solution  of  practical  problems  are  often  irra- 
tional and  require  approximating.  The  nature  of  the  problem 
must  determine  to  how  many  decimal  places  the  result  should 
be  expressed,  but  for  practice  we  shall  use  two  decimal  places. 


QUADRATIC   EQUATIONS  271 

EXAMPLE 
Solve:  ■  a;2-6.T-f  3  =  0.  (i) 

Eearranging  and  com-  ^2  —  fir4-Q  —  fi  ( <i>\ 

pleting  the  square,  X    -  Ki  x  ^  M  -Ki.      ^  (^) 

.-,  X-3  =  ±a/6._  (5) 

.-.  X  =  3  +  Ve  and  3  -  V6.       (^) 

To  test  this  work  we  must  substitute  3  +  V6  and  3  —  \/6  in  equation  (1 ). 
But,  by  taking  VG  to  the  nearest  hundredth,  or  as  2.45  these  roots  are 
3  +  v''6  =  5.45  and  3  —  \/6  =  0.55.  These  roots  will  not  check  since  they 
are  only  approximations,  but  they  are  definite  rational  numbers  and 
sufficiently  accurate  for  many  practical  purposes. 

WRITTEN    EXERCISES 

Solve,  computing  all  irrational  roots  to  two  decimal  places : 

1.  /_2/=l.  21.  x'-^x^l. 

2.  x^-\-x  =  ^.  22.  aj2  4_i0aj  =  5. 

3.  a;2  — 6a;  =  — 1.  23.  p^  —  %%,  =  —  \. 

4.  a;2  _  6  a?  =  -  3.  24.  aj^  _  g  ^j  =  13. 

5.  a;2-|-5a;  =  6.  25.  a;^  +  5  a;  =  7. 

6.  «2_4a.=  20.  26.  x'-\S)x  =  2^, 

7.  a;2  +  -i.a;  =  |.  27.  aj2-9a;  =  f. 

8.  a;2_^aj  =  i  28.  «2-|a;  =  i. 

9.  z'-A&z^-lh,  29.  22_i6;2==9. 

10.  «2_2f  =  6.  30.  ^2  _  8^  ^6^ 

11.  ?t2_^^^l^  31_  ?^2_^^5^ 

12.  a;2  +  13a;  =  -30.  32.  a)2  +  3a;  =  10. 

13.  aj2+6cc  =  -4.  33.  m2-4m  =  l. 

14.  a;2  _  8  ^  ^  _  8,  34.  ^2  +  5  w  +  6  =  0. 

15.  a;2_^13a;  =  f.  35.  ^2  +  9^4.20=0. 

16.  aj2  +  15a?  =  -25.  36.  v2_^,_2o  =  0. 

17.  a;2-8a;  =  3.  37.  a.-^  _  a;  -  42  =  0. 

18.  m2  +  8m  =  4.  38.  a;^- 5  a; -84  =  0. 

19.  x'^X^x^l.  39.  w2_|_i9^_l_84=  0. 

20.  a;2  -  5  «  +  6  =  0.  40.  ;22  _  9  ^  _l_  ^4  ^  Q 


272  A  HIGH   SCHOOL  ALGEBRA 

373.  Solution  by  Formula.  When  the  coefficient  of  x^  is  not 
unity,  the  equation  must  be  divided  by  that  coefficient  before 
using  the  method  above  to  complete  the  square. 

If  the  square  of  the  general  equation  x^-^px  +  q  =  OhQ  com- 
pleted, we  have 

x"  +  px  +^  =  ^  -  q.  Sec.  371. 

4       4 

Extracting  the  square  root  of  both  members, 

a;4-|  =  ±^£'-g,  Sec.  151. 

or,  x  =  -^±)- Vp2-4g.  Sec.  364. 

If  the  general  equation  a^  -f  6a;  -f  c  =  0  be  divided  by  a  and 

solved  as  above  the  roots  are 

^  _  -  6  ±  V&2  -  4  ac 
X  — — • 

2a 

These  results  are  general  formulas  for  finding  the  values  of 
X  in  any  quadratic  equation. 

EXAMPLE 

Solve:  5a;2  +  7a;-2=0. 

Dividing  by  5,                              CC^  +  |  cc  —  f  =  0. 
Here  p=-\-\  and  q  =  — f.  


|  =  Xand^^?34i=V|-4(-|), 

and  a:=-xVi  jVlf +f 

WRITTEN    EXERCISES 

Solve : 

1.  4a;2  +  6a;-4  =  0.  7.  4  ic^H- 12  a;-55  =  0. 

2.  9a;2  +  15a;  +  6  =  0  8.  9  m;^ +  6  w;-35  =  0. 

3.  4a;2_2a;-2=.0.  9.  9  v2_39^_^22  =  0 

4.  9a;2  +  3a;-6  =  0.  10.  4  2/^  - 12  ?/ =  91. 

5.  2522-20^  +  4  =  0.  11.  16i2_8^^i5. 

6.  3a;2-7a;-20  =  0.  12.  422  +  202  =  -21. 


QUADRATIC  EQUATIONS  273 

13.  6aj2  +  a;  =  12.  15.    12a;2  =  5a;  +  2. 

14.  6aj2  =  -5a;-f 4.  16.   9x'^  =  lSx-5. 

Clear  each  of  the  following  of  fractions  and  solve  the  result- 
ing equation : 

IT.   JL21  =  1?0_2.  23. 

a;  4- 3        x 

18      5a;       3a;  — 2_2 
a;  +  4     2a;-3~ 

19.  a;-^^  =  2. 

20.  i+     1       ■         1 


3      S  +  x     3  +  2a; 
^,     a;  +  22      4_9a;-6 
^^'   ~3  x-""^~- 

22     ^±^-ill^  =  2i 

x-l       2a;         ^'  5-a;  "  4-a;     a;  +  2' 

29.   The  perimeter  of  a  rectangular  field  is  100  yd.  and  its 
area  is  600  sq.  yd.     Find  its  length  and  breadth. 

Solution.    1.   Let  x  be  the  length  of  the  field. 

2.  Then  —  is  its  width,  and 

X 

3.  2  a:  H — '- is  its  perimeter,  being  twice  the  sum  of  its  sides. 

X 

4.  .  •.  2  cc  +  ?J_^  =  100,  by  the  conditions  of  the  problem. 

X 

5.  .-.  a;2  -  50  X  4-  600  =  0,   simplifying  (4). 

6.  .-.  (x-20)(x-30)  =  0,   factoring  (6). 

7.  .-.  a;  =  20,  ic  =  30,   solving  (6). 

8.  If  20  yd.  be  taken  as  the  length,  the  width  is  30  yd.,  by  step  (2). 

9.  If  30  yd.  be  taken  as  the  length,  the  width  is  20  yd.,  by  step  (2). 


30.  The  area  of  the  whole  plot  shown 
in  the  diagram  is  96  sq.  yd.  What  is  the 
length  of  a  side  of  the  square  (s)  ? 

31.  The  sum  of  two  unequal  sides  of  a 
rectangular  court  is  19  yd. ;  the  sum  of  the  areas  of  the  squares 


10  YD. 


274         A  HIGH  SCHOOL  ALGEBRA 

on  these  two  sides  is  181  sq.  yd.     What  are  the  dimensions  of 
the  court  ? 

32.  A  triangle  whose  area  is  200  sq.  yd.  has 
its  altitude  equal  to  its  base.  Find  the  base  of 
the  triangle,  using  the  formula,  area  =  |-  base  x 
altitude. 

33.  The  breadth  of  a  room  is  4  ft.  more  than  its  height  and 
20  ft.  less  than  its  length.  The  cost  of  cleaning  and  painting 
the  side  walls  was  $20.70,  at  2i^  per  square  foot.  Find  the 
dimensions  of  the  room. 

34.  .A  partition  is  built  5  yd.  from  one  side 
of  a  square  room  as  shown  in  the  diagram; 
the  area  of  the  floor  remaining  is  24  sq.  yd. 
What  are  the  dimensions  of  the  floor  ? 

35.  One  side  of  a  rectangle  is  f  as  long  as  the  other;  if 
10  ft.  be  added  to  the  shorter  side  and  10  ft.  be  subtracted 
from  the  longer  side,  the  area  will  not  be  changed.  Find  the 
dimensions  of  the  original  rectangle. 

36.  If  there  are  s  subscribers  in  a  telephone  exchange,  the 
total  number  of  different  connections  of  any  subscriber  with 

s(s  —  1) 
any  other  is  -^— — ^.     If  in  an  exchange  the  total  number  of 

different  connectiony  is  3240,  find  the  number  of  subscribers. 

37.  A  builder  used  steel  bars  weighing  120  lb.  each  for  a  cer- 
tain purpose.  By  changing  the  mode  of  support,  he  found  that 
he  could  get  the  same  service  from  bars  weighing  2  lb.  less  per 
running  foot,  but  2  ft.  longer  than  the  original  bars.  The  new 
bars  also  weighed  120  lb.     Find  the  length  of  the  original  bars. 

374.  Quadratic  Forms.  Certain  equations  of  higher  degree 
have  the  form  of  the  quadratic  equation  and  may  be  solved 
like  a  quadratic.     These  are  said  to  be  in  quadratic  form. 

Only  a  few  of  these  will  be  given  here  because  their  roots 
are  usually  complex  in  form  and  equations  of  this  kind  are 
given  in  Chapter  XXX. 


QUADRATIC   EQUATIONS  275 

EXAMPLES 

1.  Solve:  a:4_i3^2_j_3(3^0.  (1) 

Let  y  =  052,  then                  y^  —  13  y  +  36  =  0.  (^) 

Factoring,                              {y  -  9)  (?/  -  4)  =  0.  (3) 

Hence,                                   ?/  •  -  9  and  y  =  4.  (^) 

Since  j^  =  052                      a;2  =  9  and  x^  =  4.  (J) 

.-.  a;  =  +  3,  -  3  and  a;  =+  2,  -  2.                         (6) 

All  of  these  roots  check,  making  four  values  of  x  for  the  given  equation 
of  the  fourth  degree. 

2.  Solve:  a-^  -  9aj2  +  8  =  0.  (1) 

Let  y  =  aj2 ;  then  the  given       -,.2  _  q ,,  4.  «  =  0  (S) 

equation  becomes,  ^         ^  y  t"  '-^  •  V'^'y 

Solving  for  y,  y  =  8  OT  1.  {3) 

Therefore  X^  =  8.  {4) 

Or,  a:2  =  1.  (5) 

Solving  (4),  (5),  X  =  ±  V8,  ±  1.  (6) 

Test  these  four  values  of  x  by  substitution  in  the  given  equation. 

WRITTEN  EXERCISES 
Solve  and  test : 

1.  aj4_5^2_^4^0.  5.  R'-1^R''^S1=0. 

2.  x'-llx'  +  lQ^O.  6.  (a;-l)4-5(a^-l)2  +  4  =  0. 

3.  36a;*-13a:2 4-1  =  0.  7.  (^/-S)^- 17(2/- 5)2+16  =  0. 

4.  42/^-172/2  +  4  =  0.  8.  (^-3)4-11(^-3)2-42  =  0. 

RADICAL  EQUATIONS 

375.  To  solve  equations  in  which  only  a  single  square  root 
occurs,  transpose  so  that  the  square  root  constitutes  one  member. 
Square  both  members  and  solve  the  resulting  equation. 

EXAMPLE 

Solve:  2a;-3=Vic2^6a;-6.      (i) 

Squaring  both  members,  4  X^  —  12  X  +  9  =  X^  +  6  X  —  6.  *      {2) 

Collecting  terms,  3  X^  —  18  X  +  15  =  0.  {3) 

Solving  (5),  X  =  5  or  1.       '  (4) 

Test.  By  trial,  it  appears  that  5  satisfies  the  given  equation,  taking  the 
radical  as  positive,  while  1  satisfies  the  equation  2  x  —  3  =  —  Vx'-^  -|-  0  x  —  (J. 


276         A  HIGH  SCHOOL  ALGEBRA 

1.  It  must  be  remembered  that  the  equation  resulting  from  squaring 
will  usually  not  be  equivalent  to  the  given  equation  (Sec.  172,  p.  128). 
It  may  have  additional  roots,  and  substitution  must  determine  which  of 
the  values  found  satisfy  the  given  equation. 

2.  In  order  that  the  given  problem  may  be  definite,  the  radical  must 
be  taken  with  a  given  sign.  If  every  possible  square  root  is  meant,  two 
different  equations  are  really  given  for  solution.  Thus,  unless  restricted, 
2x  =  \/4  —  6 X  is  a  compact  way  of  uniting  the  two  different  equations, 
2x  =  +  \/4  —  6  X,  and  2  cc  =  —  V4  —  i5x.  If  solved  as  indicated  above, 
it  appears  that  the  first  is  satisfied  when  x  =  ^,  the  second  when  x  =—  2. 

3.  In  the  exercises  of  the  following  set  the  radical  sign  is  to  be  under- 
stood to  mean  the  positive  square  root. 

376.  Extraneous  Roots.  The  ambiguity  of  signs  is  removed 
by  squaring;  hence,  the  resulting  equation  appears  to  have 
more  roots  than  the  original.  These  roots  of  the  final  equation 
which  do  not  satisfy  the  original  are  called  extraneous  roots, 
and  should  be  omitted. 


WRITTEN    EXERCISES 
Solve,  and  test  only  rational  roots : 


2.    x  =  Vbi-x-bx.  10.   30=a;-29Va;. 


3. 

3a;-7Va;  =  -2. 

4. 
5. 

Va;  -1-  4  —  a;  =  4. 

V^  — 1       X 

3           16 

6. 

a;  +  5V37-aj  =  43. 

7. 

^x-\-5  =  x-7. 

8. 

V2a;  +  7  =  |. 

11.   ic  =  2+V3-lla;. 


12.    a;  —  Va;  —  2  =  2. 


13.    »4-l  =  Va;  +  3. 


14.    VlOO-a;2=40 


15.    x-\--V2x—x^  =  2. 

_     x-{-2     x-hl      V2 a;  +  1  _  A 
''•    "2 3  2        -^' 

377.  If  the  equation  involves  but  one  radical,  the  method 
of  Sec.  375  can  be  used ;  but  when  more  than  one  radical  occurs, 
successive  squaring  may  be  necessary. 


QUADRATIC   EQUATIONS  277 

EXAMPLE 


Solve:     V2a;  +  6  +  V3aj4-l  =  8.  (i) 

Rearranging,                     ^/2x  +  6  =  8  -  VS  X  +  1-  (^) 

Squaring,                          2  a;  +  6  =  64  -  16  VS  x+l+Sx  +  1.  (3) 

Collecting,                    16  \/3  X  +  1  =  x  +  59.  (4) 

Squaring  again    a;2_ 650  X  + 3225=0.  (5) 

and  collecting,  ^    ^ 

Solving,                                     X  =  5,  or  645.  (6) 

Test.  Trial  shows  that  the  first  of  these  values  satisfies  the  given  equa- 
tion, and  it  is  obvious  by  inspection  that  the  second  cannot  satisfy  the 
equation. 

378.  Sometimes  it  is  best  first  to  transform  the  given  ex- 
pression. 

EXAMPLE 


2x-\-l-\-V5x-{-6 


Solve:  2x-\-^4.x^  +  9  =  -^'^~^^^  v^u^^r^,  ^^. 


Clearing  of  fractions,  9  =  2  X  +  1  +  \/5  X  +  6.  (^) 

Rearranging,  8  —  2  X  =  VS  X  +  6.  (3) 

Hence,  X  =  2,  or  7|.  (5) 

Test.  By  trial,  2  is  seen  to  satisfy  the  given  equation.  To  avoid  the 
complete  work  of  substituting  1^,  we  note  that  every  root  of  (1)  must 
satisfy  (3).  If  x  =  7J,  the  left  member  of  (3)  is  negative  and  the  right 
member  positive.  Hence,  7^  is  an  extraneous  root  (Sec.  376)  and  should 
be  disregarded.     It  would  satisfy  (3)  if  the  radical  had  the  negative  sign. 


WRITTEN    EXERCISES 


Solve 


2.    V^  — Va;  — 15  =  1. 
6 


3.  Va;2  -  5  H — =  5. 

Va;2-5 

4.  x-}-2-\-(x-{-2y  =  20. 

5.  V^"qp7-4-V3a;-2=-A^+J-. 

V3.r-2 


278  A  HIGH  SCHOOL   ALGEBRA 

6.  Wx'^-{-5x-j-6  =  V5  .  V«2 -\-5x-\-6. 

7.  -Vx^  -  a'x^  =  Vx*  +  64a;2. 

8.  Vx  H-  Va  —  a;  =  Va  +  6. 


9.    V6^  +  6-V3;2  +  l=V52-21. 
10 


10.    V44-2+-^  =  V5 


379.   An  equation  whose  radicals   have  been   removed  by 
squaring  may  be  linear. 


WRITTEN    EXERCISES 

Solve  and  test : 


1.  Vy  -  Vy  -  8  =  3.   Vic  - 1  -  Va  - 1  =  0. 

-Vy-S 

2  .1.1  1 


2.    Vic  — 1  — Va;  =  — ^. 


Va;  1— a;     i_j_ya;     1— Va; 

REVIEW 
WRITTEN    EXERCISES 

Solve,  approximating  any  irrational  roots  to  two  decimal 
places : 

1.  a;2  +  7a;  =  8.  11.   322-2^  =  1. 

2.  3a;2  =  48.  12.   y^ +  4:  ay  -2  =  0. 

3.  a;2  +  25a;  =  -100.  13.    t^-^3t-6  =  0. 

4.  (3  a; +  4)2=  96.  14.   13  aj^  -  39  a;  =  0. 

5.  aj2-25a;+144=0.  15.    aV  -  a6a;  =  0. 

6.  a;2  +  3a;-28  =  0.  16.    (a -\- b)x'^ -{- ex  =  0. 

7.  a;2  — 13a;  =  68.  17.   mV  -  (m  +  r)a;  =  0. 

8.  a;2-12a;  +  27  =  0.  18.    ax""  -  bx  =  cx'^  +  dx. 

9.  a;2  _|.  Ill  aj=:  3400.  19.   ix^  =  U-3x\ 
10.    5  a;2  + 13  a;  =  370.  20.   a;^  +  5  =  i^  a;^  -  16. 


QUADRATIC   EQUATIONS  279 


21.  M_M  =  ,  +  i. 


28. 

12x^-Sx  =  7x''-\-2x. 

29. 

10a^  +  5a;  =  15a;2  4.9a;. 

30. 

(a+6)V+(a  +  6)a;  =  0. 

31. 

^-{-4:=2X. 

X 

22.  x-2-\-V2-x  =  0. 

23.  a;2 +  130-0;  =  19. 

24.  5-3a;  +  la;2  =  0.  ,^ , 

'^  32.   a;4-V9-a;2=4. 

25.  a;2-7a;  =  0.  33    ^4.10^,2^9^0. 

26.  y'-ay^c.  34.    (2_3)4_5(2_3)  =  _4. 

27.  aj2_i2  =  30+a;.  35.    (w  -  7)*  -  (w;  -  7)^  =  2. 

36.  The  perimeter  of  a  rectangular  field  is  200  ft.,  and  its 
area  is  2400  sq.  ft.     Find  its  length  and  breadth. 

37.  The  area  of  a  rectangular  field  is  2000  sq.  ft.,  and  its 
length  is  10  ft.  more  than  its  breadth.     Find  its  dimensions. 

38.  After  a  man  had  lived  in  his  house  4  months  longer 
than  the  number  of  dollars  monthly  rental  that  he  paid  for  his 
house,  he  had  paid  altogether  $  320  rent.  How  much  was  the 
monthly  rental  ? 

39.  If  d  is  the  diagonal  of  a  square  of  side  s,  then  d^  =  2  s^. 
Solve  this  equation  for  d.     For  s. 

40.  In  Exercise  39,  when  s  =  4,  find  d,  taking  1.414  as  the 
square  root  of  2. 

41.  The  volume  (v)  of  a  cylinder  is  the  product  of  the  area 
of  the  base  (irr^)  and  the  height  (li).  That  is,  v  =  trr'^h.  Solve 
this  equation  for  r.     For  h. 

42.  Using  -2y2-  for  tt  in  v  =  Trr^/i,  find  the  radius  of  the  base 
of  a  cylinder,  if  its  volume  is  -^  sq.  ft.  and  its  altitude  4  ft. 

43.  Given  m  =^.     Express  v  in  terms  of  the  other  letters. 

44.  In  the  preceding  exercise,  express  c  in  terms  of  the 
other  letters. 

45.  A  room  is  1  yd.  longer  than  it  is  wide;  at  75^  per 
square  yard,  a  covering  for  the  floor  of  the  room  costs  $  31.50. 
Find  the  dimensions  of  the  floor. 

19 


280  A   HIGH  SCHOOL  ALGEBRA 

46.  The  length  of  a  room  is  twice  its  height,  and  the  breadth 
is  6  ft.  more  than  the  height.  At  10^  per  square  foot  it  costs 
$72  to  decorate  the  side  walls  of  the  room,  no  allowance  being 
made  for  openings.     Find  the  dimensions  of  the  room. 

47.  The  cost  of  decorating  a  certain  square  ceiling  is  $45. 
If  a  second  square  ceiling  of  side  5  yd.  longer  were  decorated 
at  the  same  rate,  the  cost  would  be  $  80.  Find  the  dimensions 
of  the  first  ceiling. 

48.  A  hall  is  lighted  by  a  certain  number  of  incandescent 
electric  lights  and  5  fewer  of  gas  mantles.  The  candle  power 
of  each  of  the  former  is  70  greater  than  that  of  each  mantle. 
The  total  candle  power  of  the  gas  mantles  is  500,  and  that  of 
the  electric  lights  1800.     How  many  lamps  of  each  sort  ? 

SUMMARY 

The  following  questions  summarize  the  definitions  and  pro- 
cesses treated  in  this  chapter  : 

1.  Define  and  illustrate  a  complete  quadratic  equation;  also 
an  incomplete  quadratic  equation.  Sec.  369. 

2.  What  is  apwre  quadratic^    An  affected  quadratic? 

Sec.  370. 

3.  Explain  the  process  of  completing  the  square.      Sec.  371. 

4.  Why  approximate  roots?  Why  not  use  such  roots  for 
testing  ?  Sec.  372. 

5.  Explain  how  to  proceed  when  the  coefficient  of  x^  is  not 
unity.  "  Sec.  373. 

6.  State  th.Q  formula  for  solving  quadratic  equations. 

Sec.  373. 

7.  What  kind  of  higher  equations  may  be  solved  by  quad- 
ratic methods  ?  Sec.  374. 

8.  How  are  radical  equations  solved  ? 

Sees.  375,  377,  378. 

9.  Define  extraneous  roots. 

Sec.  376. 


QUADRATIC   EQUATIONS  281 


HISTORICAL  NOTE 

The  solution  of  quadratic  equations  dates  from  Diophantos,  but  he  did 
not  leave  any  description  of  a  general  method.  The  Hindoos  could  solve 
special  cases  by  completing  the  square  when  the  number  to  be  added  for 
this  purpose  was  obvious,  and  Cridharra  is  said  to  have  formulated  a  rule 
for  this.  They  discussed  the  existence  of  two  roots,  although  they  felt 
that  positive  roots  only  were  significant,  and  Bhaskara  called  a  negative 
root  "  inadequate,  because  people  don't  approve  of  negative  roots."  The 
Arab,  Mohammed  ben  Musa,  or  Al-Khowarazmi  (p.  93),  helped  to 
simplify  the  solution  of  equations  by  giving  directions  "  first  to  transpose 
terms,  then  combine  them,"  but  added  nothing  else  of  importance. 

The  European  scholars  of  the  Middle  Ages  contented  themselves  with 
translating  the  Greek  and  the  Arab  manuscripts  so  that  further  develc^- 
ment  in  solving  equations  awaited  the  mathematicians  of  the  fifteenth 
century.  By  that  time  there  were  current  more  than  twenty  special  rules 
for  solving  quadratics,  which  Michael  Stifel  (about  1550)  reduced  to  three. 

Stifel,  the  greatest  German  algebraist  of  the  sixteenth  century,  was 
born  in  Esslingen  and  was  educated  by  the  monks  for  the  ministry  ;  but 
his  interest  in  the  mystic  numbers  found  in  the  prophetic  books  of  the 
Bible  led  him  to  study  mathematics.  The  new  science  of  algebra  was 
called  by  the  Germans  "  Coss,"  and  Stifel  is  now  known  as  the  greatest 
"cossist."  He  made  many  improvements  in  algebra,  but  his  limited 
idea  of  the  negative  number  hindered  him,  particularly  in  the  solution 
of  equations.  Thus,  he  succeeded  only  in  decreasing  the  number  of  rules 
for  solving  the  quadratic  equation,  whereas  Stevin,  in  the  next  century, 
with  full  knowledge  of  the  negative  number,  reduced  all  of  these  rules  to 
our  present  method  of  completing  the  square. 


CHAPTER   XXIII 
SYSTEMS  OF   QUADRATIC   EQUATIONS 
SIMULTANEOUS   QUADRATIC  EQUATIONS 

380.  Two  simultaneous  quadratic  equations  with  two  un- 
knowns cannot  in  general  be  solved  by  the  methods  used  in 
solving  quadratic  equations,  because  an  equation  of  higher 
decree  usually  results  from  eliminating  one  of  the  unknowns. 
But  many  systems  containing  quadratic  equations  can  be  solved 
by  quadratic  methods,  among  them  the  following : 

381.  Class  I.  In  which  the  result  of  substitution  is  a  quad- 
ratic form. 

1.  A  system  of  equations  composed  of  a  linear  equation  and  a 
quadratic  equation  can  always  he  solved  by  substitution. 


EXAMPLE 


Solve : 


From  (i), 
From  (3), 

Substituting  (4)  in  (^), 

Simplifying  (5), 
Factoring  (6), 

Solving  (7), 
From  (4), 


Test. 


3  .  4  +  4  .  3  =  24 

32  +  42  =  25 


3  x  +  4  ?/  =  24. 

a;2  +  2/2  =  25. 

3  .r  =  24  —  4  y. 


r 


25. 


and 


25  ?/2  -I92y  +  351  =  0. 
(y-3)(25i/-117)=0. 

y  =  3andy=l^ 
25 

44 

x  =  4  and  x  =  — 

3-44  ,  4  .  117 
25 


+ 


25 


24. 


I 
282 


iM^m=^^' 


{2) 
(3) 

(6) 

(7) 

(9) 


SYSTEMS  OF  QUADRATIC   EQUATIONS  283 

2.  When  both  equations  are  quadratic,  substitution  is  appli- 
cable if  the  result  of  substitution  is  an  equation  having  the  quad- 
ratic form. 


Solve: 


EXAMPLE 

lx'-\-y^  =  25,  (i) 

I         xy  =  12.  (2) 

From  (2),                                                                        X  =  — .  ,            (5) 


Substituting  from  (5)  in  (J), 


Simplifying  (U),  we  obtain  an  4  _  05  ,.2  4.  144  _  Q  r^^ 

equation  in  quadratic  form,        ^         ^^  y    -r  J-tt  —  v.  \,0) 

Factoring  (5),  (y2  _}.  16)  (j/^  _  9)  =  Q.  (6) 

Solving  (6) ,  '  y  =  ±  4,  and  ±  3.      (7) 

Substituting  (7)  in  (3),  iC  =  ±  3,  and  ±  4.       {8) 

Test.    Taking  both  values  to  be  positive,  or  both  to  be  negative, 

(±3)2 +  (±4)2  =  25. 
(±3).  ±4  =  12. 

There  are  frequently  various  methods  of  solving  the  same  problem. 
Thus,  in  the  last  example,  multiply  {2)  by  2,  add  it  to  or  subtract  it  from 
(i);  the  resulting  equations  (x  +  ?/)2  =  49,  and  (a;  —  y)2  =  1,  can  be 
solved  by  extracting  the  square  roots  of  both  members  and  adding  and 
subtracting  the  resulting  equations. 

382.  Corresponding  Values.  The  sets  of  values  of  x  and  y 
which  form  solutions  may  be  determined  by  noticing  which 
value  of  one  unknown  furnishes  a  given  value  of  the  other  in 
the  process  of  solution. 

Thus,  in  Example  1,  p.  282,  ?/  =  3  produces  x  =  4. 

WRITTEN    EXERCISES 


Solve  and  test : 

. 

1.    a;2  +  2/2  =  2  a;?/,  ' 

4. 

x-y  =  l, 

7. 

a;  +  2/  =  a, 

x-\-y=^. 

a;2  _  /  =  16. 

a;2-2/2=6. 

2.   a;2  +  2/2=:25, 

5. 

^^2  _  ,,2  ^  1(3^ 

8. 

x-y  =  b, 

x-\-y  =  l. 

m—  n  —  2. 

xy  =  a\ 

3.    a;2  +  2/'  =  13, 

6. 

2m-R^^=U, 

9. 

2k  =  k^~  k'% 

2  X  -f-  3  2/  =  13. 

3i2  +  i?i  =  ll. 

2k  =  4  kk'. 

284  A  HIGH  SCHOOL  ALGEBRA 

383.    Class  II.     In  which  one  equation  is  homogeneous. 

A  system  in  which  one  equation  has  only  terms  of  the  second 
degree  in  x  and  y  can  he  solved  by  finding  x  in  terms  of  j,  or 
vice  versa,  from  this  equation  and  substituting  in  the  other. 


Solve : 


EXAMPLE 

x''-5xy  +  6?/2  =  0,  (i) 

aj2  _  y2  ^  27.  (^) 

Dividing  (1)  by  yi,  —  -  ^  +  ^  =  0.  (3) 

y2  y2  y-2 


Simplifying  (3), 
Factoring  (4), 


(^)^-5(-^)  +  6  =  0.  (,) 

(^^)(^3)=0.  (5) 

Solving  (5),  ^  =  2,- =3.  (6) 

y        y 

Froni  (6),  x  =  2y,  x  =  Sy.  (7) 

Substituting  a?  =  2  y  in  (2),  (2  y)^  -  y^  —  27.  {8) 

Solving  (8),  2/  =  ±  3.  (5) 

Substituting  (9)  in  a?  =  2  y,  X=±Q.  (10) 

Substituting  x  =  Sym  (2),  (3  yY  -  y'^  =  27.  (11) 

Solving  (li),  1/  =  -^  =  ±  I V6.     (i^) 

2\/6 

4-27  - 

Substituting  (I^)  in  aj  =  3  y,  X  =  ^—^  =  ±W6.    (13) 

2V6 

Test. 

Taking  both  values  to    f  (  ±  6)2-  5(  ±  3)  (  ±  6)  +  6(  ±  3)^  =  0. 


iking  both  values  to    f  ( -j.  6)2—  5(  ±  3)  ( ±  € 
S:gffvr'''''M(±6)2-(±3)2  =  27 

V2V6/  \2V6/\2V0/  V2V6/ 

f±^Y^_f^y^.27. 
\2V6/       \2\/6/ 


Taking  the  signs  as 
before, 


Notes.     1.   When  the  equation  in  -,  as  in  step  (4),  cannot  be  factored 

y 

by  inspection,  the  formula  for  solving  the  quadratic  equation  is  used, 

2.  When  an  equation  has  the  right  member  zero,  it  is  unnecessary  to 
divide  by  x2  or  ?/2,  if  the  left  member  can  be  factored  by  inspection. 
Thus,  in  (1)  above,  (x  —  3  ?/) (x  -  2  2/)  =  0,  hence  x  =  ^y  and  a:  =  2 ?/,  as 
in  (7). 


SYSTEMS  OF  QUADRATIC   EQUATIONS  285 

WRITTEN   EXERCISES 

Solve : 

1.  x'^  —  2xy  —  Sy^  =  0,  6.  6  x^ -{-  5xy  +  y^  =  0, 
a;2  +  2  2/2  =  12.  y2_x-y  =  32. 

2.  x'^-\-xy-2y^  =  0,  7.  3t^-2  tu  -  u"^  =  0^ 
x^  +  y^  =  50.  t  i-u  +  u''  =  32. 

3.  x'^-y'^  =  0,  8.    x''-\-ay  =  0, 

x"^  —  S  xy  -{-  2  y  =  I.  x^  -{-  ay  —  y"^  =  a?. 

^.  2x^  —  5xy  -\-2y^  =  0,  9.  x'^  -{-  S  x  —  5  y  -\-  xy  =  400, 

4  aj2  -  4  2/2  +  12  a;  =  3.  x^ -\- 7  xy  +  10  y^  =  0. 

5.  m(m  4-  w)=  0,  10.  4  aj^  +  4  a^  +  2/^  =  0, 

m2  -  mn  +  ^2  =  27.  x^-{-3y^  —  2x  =  195. 

384.  Class  III.  In  which  the  imknowns  are  symmetrically 
involved. 

A  system  in  tchich  each  equation  is  unaltered  ivhen  x  and  j  are 
interchanged  can  be  solved  by  letting  x  =  u  -f  v  and  y  =  u  —  v. 

EXAMPLE 

Solve:  I         ^'  +  f=^^  « 

xy-{-x  +  y  =  5.  (2) 

'Letx=u  +  v,  y—u  —  v, 

then  (J)  becomes     (u  +  v)^  +  (m  —  v)^  =  5,  (5) 

and  (2)  becomes 

(u-]-v)(u  —  v)  +  (ic-{-v)  +  (u-v)  =  6.  (4) 

Simplifying  (3),                                 2  U^  +  2  V^  =  5.  (5) 

Simplifying  (U),                            U^  -  v'^  -\- 2  U  =  5.  (6) 

Dividing    (5)    by  2,  and  2u^-l-2u—i^  (7\ 

subtracting  it  from  (6),  ^u    -^  ^u  —   ■^-.  \^/ ) 

Simplifying,  ^2  _^  ^  _  J^i  _  0.  (5) 

^    Solving  (S),  W=-^±iVl  +  15,  (P) 

Substituting  in  (5),  2  .  f  +  2  V^  =  5.  (iO) 

Solving  (io),  'uS  _  1^  and«=— I,  +|.  (ii) 

Finally,  X  =  !*  +  V  =  1^  q=  ^  =  1  or  2,  (i;2) 

and,  y  =  u  —  v  =  \\±\=2  or\.  {13) 

If  tHe  other  value,  m  =—  2|,  step  (5),  be  substituted  in  (5),  then 
V  =  ±  ^  V—  15.  But  square  roots  of  negative  numbers,  called  imaginaries, 
will  not  be  admitted  until  they  have  been  studied  in  Chapter  XXVIII. 


286  A  HIGH  SCHOOL  ALGEBRA 

Special  systems  of  symmetric  equations  may  be  solved  by 
dividing  one  equation  by  the  other. 

For  example, 

x^  +  y^  =  18,     ^j,     lx^  +  a;2y2  +  ^i  ^ 


a;  +  ?/ =  6,  [x^  —  xy  +  y^  =  4. 

WRITTEN    EXERCISES 

Solve  by  Sec.  384  or  by  some  of  the  previous  methods : 


1. 

x-y-\-l  =  0. 

7. 

xy  =  S(x-\-y), 
x'  +  f-^  160. 

2. 

x'-xy  +  f-  =  12, 
a.'2  +  xi/  +  2/'  =  4. 

8. 

X     y_^             . • 
V     a;     2 

3. 
4. 

xy—{x-\-y)-l=0, 
xy  =  -S. 
a^  +  2/'  =  39, 
y-x  =  3. 

9. 

1  +  1  =  4. 
X     y 

x'  +  y'  +  x  +  y  =  l%^, 
xy  =  77. 

5. 

2x'^  +  2f-(x-y)=:9, 
2/^  =  1. 

10. 

x-y  =  3. 

6. 

x^-xy-\-y'=:19, 
xy  =  15. 

11. 

x'-^afy'-^y'  =  S, 
x'  +  xy  +  f  =  2. 

385.  The  foregoing  classes  of  simultaneous  quadratic  equa- 
tions are  applied  in  the  following  problems. 

WRITTEN    EXERCISES 

1.  Two  square  floors  are  paved  with  stones  1  ft.  square ; 
the  length  of  the  side  of  one  floor  is  12  ft.  more  than  that  of 
the  other,  and  the  number  of  stones  in  the  two  floors  is  2120. 
Find  the  length  of  the  side  of  each  floor. 

Suggp:stion.  Let  x  be  the  length  in  feet  of  a  side  of  the  smaller  floor 
and  y  be  the  length  of  the  other,  then 

x  =  y-\2,  (i) 

and  by  the  given  conditions,  X^  +  y'^  —  2120.  {2) 

Substituting  (I)  in  {2),  {y  —  12)2  _(.  y2  -  2120.  {3) 

Simplifying  and  factoring  (3),  {y  —  38)  {y  -f  26)  =  0  (4) 

The  negative  values  not  being  admissible,  the  squares  are  26  ft.  and  38  ft. 
on  a  side. 


SYSTEMS   OF   QUADRATIC   EQUATIONS  287 

2.  The  sum  of  the  sides  of  two  squares  is  7  and  the  sum  of 
their  areas  is  25.     Find  the  side  of  each  square. 

3.  The  hypotenuse  of  a  certain  right  triangle  is  50,  and  the 
length  of  one  of  its  sides  is  f  that  of  the  other.     Find  the  sides. 

4.  The  difference  between  the  hypotenuse  of  a  right  triangle 
and  the  other  two  sides  is  3  and  6  respectively.     Find  the  sides. 

5.  A  number  consists  of  two  digits ;  the  sum  of  their 
squares  is  41.  If  each  digit  is  multiplied  by  5,  the  sum  of 
these  products  is  equal  to  the  number.     Find  the  number. 

6.  The  difference  between  two  numbers  is  5 ;  their  product 
exceeds  their  sum  by  13.     Find  the  numbers. 

7.  The  diagonal  of  a  rectangle  is  13  in. ;  the  difference  be- 
tween its  sides  is  7  in.     Find  the  sides. 

8.  The  diagonal  of  a  rectangle  is  29  yd.,  and  the  sum  of  its 
sides  is  41  yd.     Find  the  sides. 

9.  The  sum  of  the  perimeters  of  two  squares  is  104  ft. ;  the 
sum  of  their  areas  is  346  sq.  ft.     Find  their  sides. 

10.  The  difference  between  the  areas  of  two  squares  is  231 
sq.  in. ;  the  difference  between  their  perimeters  is  28  in.  Find 
their  sides. 

11.  Two  trains  leave  Xew  York  simultaneously  for  St. 
Louis,  which  is  1170  mi.  distant ;  the  one  goes  10  mi.  per  hour 
faster  than  the  other  and  arrives  9J  hr.  sooner.  Find  the  rate 
of  each  train. 

12.  In  going  120  yd.  the  front  wheel  of  a  wagon  makes  6 
revolutions  more  than  the  rear  wheel ;  but  if  the  circumference 
of  each  wheel  were  increased  3  ft.,  the  front  wheel  would  make 
only  4  revolutions  more  than  the  rear  w^heel  in  going  the  same 
distance.     Find  the  circumference  of  each  wheel. 

13.  The  sum  of  the  volumes  of  two  cubes  is  35  cu.  in.  and  the 
sum  of  the  lengths  of  their  edges  is  5  in.  Find  the  length  of 
the  edge  of  each. 

14.  The  difference  between  the  volumes  of  two  cubes  is  37 
cu.  ft.  and  the  difference  between  their  edges  is  1  ft.  Find 
their  edges. 


288  A  HIGH  SCHOOL  ALGEBRA 

386.   Higher  Equations.     Certain  systems  containing  higher 
equations  can  be  solved  by  the  methods  of  this  chapter. 


EXAMPLES 

1.    Solve: 

'  X  +2/  =12. 

{1) 
{2) 

Put  x=  u  +  v,  and 

y  =  n-  V, 

Then,  from  (i) 

(w  +  vy  +  (w  -  vY  =  18(U  +  t7)(w  -  v), 

(3) 

and  from  {2), 

(u  +  v)  +  (u-v)=2u  =  12. 
.-.  M  =  6. 

(4) 

Combining  (4)  and 

(3),              216  4-18t?2  =  9(36-tj2). 

(5) 

Solving  (5), 

i)=±2. 

(6) 

From  (4)  and  (6), 

cc  =  M  +  u  =  8  and  4, 

(7) 

and 

y  =  u  —  V  =  4:  and  8. 

(S) 

Test  as  usual. 

2.    Solve: 

Put  x=  u  +  v,  '!/  = 

0:^4-^4  =  706, 
>;  -y  =2. 

(2) 

Then  from  (I), 

(^u  +  vY-\-{u-vY  =  im, 

(3) 

and  from  (^) , 

(w  +  v)-(w-v)=2. 

U) 

Simplifying  (U), 

^  =  1. 

(5) 

From  (5)  and  (3), 

(w  + 1)4+0/- 1)4  =  706. 

(0 

Simplifying  (6), 

w*  +  6  ?*2  _  352  =  0. 

(7) 

Solving  (7), 

w2  =  16, 

(<?) 

and 

w2=_22. 

(9) 

From  (S), 

u  =  ±4.. 

(10) 

From  (5)  and  (10), 

x  =  ±4  +  l  =  5,  -3. 

{11) 

and 

y=±4-l=-5,  +3. 

{12) 

The  imaginary  values  of  u  in  (9),  namely  ±  V—  22,  would  give  other 
values  of  x  and  y,  but  such  values  will  be  omitted  here  and  considered  in 
Chapter  XXVIH. 

Test  as  usual. 

WRITTEN    EXERCISES 

Solve: 

1.  a;3  +  2/3  =  189,  4.   x^  +  y^  =  2, 
x-\-y  =  ^.  a;  +  2/  =  0. 

2.  a^4-2/^=72,  5.   ar'-fr  =  2, 
x-\-y  =  Q.  x  +  y  =  2. 

3.  a^  + 2/^  =  189,  6.   a;^  +  2/*  =  17, 
0^2/  +  ^2/^  =  180.  a;  —  2/  =  1. 


SYSTEMS  OF   QUADRATIC   EQUATIONS  289 


REVIEW 

WRITTEN 

EXERCISES 

Solve : 

1.   a; +  2/ =  7.5, 

12. 

x^y  =  5, 

xy  =  14. 
2.   Sx-2y=0, 

x^y     6 

xy  =  13.5. 

13. 

x-y  =  -S, 

3.    x  +  y  =  7, 

x^-f  =  -9. 

x'-f=.21. 
4:.   x  —  y  =  5, 

14. 

x-y  =  S. 

5.   x-y  =  l, 

15. 

^-1  =  0, 

3a.-2  +  2/^  =  31. 
6.   x-y=:5, 

2     3 

it'2  +  2/2  =  5(aj  +  2/)  +  2. 

x'-\-2xyi-y'. 

=  75. 

16. 

aj  +  2/  =  9, 

7.    x  +  y=7{x- 

y), 

xy   _X0 

3^  +  2/2  =  225. 

Vxy      V5 

8.    5(x^-f)=4. 

(^4-2/'), 

17. 

ar^- 2/3^565^ 

a;  +  2/  =  8.    - 

x-y  =  5. 

9.    "^y  =4, 

x-h5y 

18. 

x^y=='^  =  x'-y\ 

xy  =  20. 

y 

10.    x^  -f  ir.?/  +  2/^  = 
xy  =  6. 

:19, 

19. 

x-{-y  =  a, 
a^  +  f  =  b\ 

11.    (a^  +  2)(aj-3; 

1=0, 

20. 

x-y  =  a, 

ar^  +  3a^2/  +  2/'^ 

=  5. 

a^-f  =  b\ 

SUMMARY 
The   following    questions    summarize    the    definitions    and 
processes  treated  in  this  chapter: 

1.  Name  three  classes  of  simultaneous  quadratic   equations 
that  can  be  solved  by  quadratic  methods.     Sees.  381,  383,  384. 

2.  State  how  the  first  class  may  be  solved ;  also  the  second 
class;  the  third  class.  Sees.  381,  383,  384. 


290  A  HIGH  SCHOOL  ALGEBRA 

3.  How  is  it  possible  to  determine  what  values  of  one  un- 
known are  to  be  associated  with  those  of  the  other  ?     Sec.  382. 

4.  State  a  set  of  higher  equations  that  can  be  solved  by 
quadratic  methods.  Sec.  386. 

HISTORICAL  NOTE 

Simultaneous  quadratic  equations  in  determinate  systems,  such  as  we 
study  in  elementary . algebra,  received  little  attention  until  recent  times. 
Mathematicians  of  the  past  favored  the  study  of  indeterminate  systems, 
that  is,  systems  in  which  there  are  more  unknown  quantities  than  equa- 
tions. These  problems  offered  a  wide  range  for  ingenuity,  and  Diophantos 
invented  many  special  devices  for  their  solution.  The  Hindoos  and  Arabs 
were  likewise  attracted  to  this  class  of  problems.  Thus,  no  real  progress 
was  made  by  Eastern  scholars  in  finding  general  solutions  of  simultaneous 
quadratic  equations. 

To  illustrate  how  little  was  known  about  this  subject  in  the  middle 
ages,  we  may  refer  to  Gerbert  (990)  who,  while  teaching  at  Rheims, 
gained  much  fame  as  a  mathematician  by  solving  the  problem  :  To  find 
the  sides  of  a  right-angled  triangle  given  its  area  and  hypotenuse.  It 
seems  strange,  from  our  present  day  point  of  view,  that  the  solution  of 
so  simple  a  pair  of  equations  as: 

(1)   x''  +  y^  =  h^  (2)    lxy  =  a 

should  have  brought  great  distinction  to  any  one.  Gerbert  is  often 
referred  to  as  Sylvester,  because  he  later  became  Pope  under  the  name 
of  Sylvester  II. 


CHAPTER   XXIV 

REVIEW  AND   EXTENSION  OF   PROCESSES 

POSITIVE  AND  NEGATIVE  NUMBERS 

387.  Algebra  is  concerned  with  the  study  of  numbers.  The 
number  of  objects  in  any  set  (for  example,  the  number  of 
books  on  a  shelf)  is  found  by  counting.  Such  numbers  are 
called  whole  numbers  or  integers ;  also  primitive  or  absolute 
numbers. 

In  arithmetic,  numbers  are  usually  represented  by  means  of 
the  numerals,  0,  1,  2,  3  ...  9,  according  to  a  system  known  as 
the  decimal  notation.  In  algebra,  numbers  are  represented  by 
numerals  and  also  by  letters,  either  singly  or  in  combinations. 

388.  Graphical  Representation.  The  natural  integers  may 
be  represented  by  equidistant  points  of  a  straight  line,  thus : 


389.  Addition.  If  two  sets  of  objects  are  united  into  a 
single  set  (for  example,  the  books  on  two  shelves  placed  on  a 
single  shelf),  the  number  of  objects  in  the  single  set  is  called 
the  sum  of  the  numbers  of  objects  in  the  two  original  sets. 
The  process  of  finding  the  sum  is  called  addition. 

The  sign,  +,  between  two  number  symbols  indicates  that 
the  numbers  are  to  be  added.  In  the  simplest  instances  the 
sum  is  found  by  counting. 

Thus,  to  find  5  +  7,  we  first  count  5,  and  then  count  7  more  of  the 
number  words  next  following  (six,  seven,  eight,  etc.).  The  number 
word  with  which  we  end  (twelve)  names  the  sum. 

291 


292         A  HIGH  SCHOOL  ALGEBRA 

390.   Graphical  Representation.     The   sum   of  two   integers 
may  be  represented  graphically  thus : 

3  +  5  a   +  b 


I — 1 — I — I — 1 — I — 1 — I — I 

012345678 

Theoretically,  the  sum  of  two  integers  can  in  every  instance  be  found 
by  counting.  But  it  is  not  necessary  or  desirable  to  do  so  when  either 
(or  both)  of  the  numbers  is  larger  than  nine.  In  this  case,  the  properties 
of  the  decimal  notation,  as  learned  in  arithmetic,  enable  us  to  abridge 
the  process  of  counting. 

391.  Commutative  Law  of  Addition.  If  two  sets  of  objects 
are  to  be  united  into  a  single  set,  the  number  of  objects  in  the 
result  is  obviously  the  same  whether  the  objects  of  the  second 
set  are  united  with  those  of  the  firstj  or  those  of  the  first  united 
with  those  of  the  second. 

For  example,  the  number  of  books  is  the  same  whether  those  on  the 
first  shelf  be  placed  on  the  second,  or  those  on  the  second  be  placed  on 
the  first. 

In  symbols  :  a  -j-  6  =  5  -f-  a. 

This  fact  is  called  the  commutative  law  of  addition. 

The  letters  a  and  b  are  here  used  to  stand  for  integers,  but  the  law  will 
apply  when  they  stand  for  any  algebraic  numbers. 

392.  Graphical  Representation.  The  commutative  law  may 
be  represented  graphically  thus  : 


393.  Addition  of  Two  or  More  Whole  Numbers.  If  more 
than  two  sets  of  objects  are  united  into  a  single  set,  the  number 
of  objects  in  the  resulting  set  is  called  the  sum  of  the  number 
of  objects  in  the  original  sets,  and  the  process  of  finding  the 
sum  is  called  addition.  As  in  the  case  of  two  numbers,  the  sum 
of  three  or  more  numbers  may  be  found  by  counting  in  the 
simplest  instances,  and  for  larger  numbers,  the  process  may  be 
abridged  by  use  of  the  properties  of  the  decimal  notation. 


REVIEW   AND   EXTENSION   OF  PROCESSES       293 

394.  The  comirnitative  law  likewise  applies  to  the  sum  of 
three  or  more  integers.     That  is  : 

Tlie  sum  is  the  same  for  every  order  of  adding  the  numbers. 

395.  Associative  Law  of  Addition.  If  we  have  three  rows  of 
books,  the  number  of  books  is  the  same  whether  those  in  the 
second  row  are  first  placed  with  the  first  row,  and  then  those  in 
the  third  row  placed  with  these,  or  those  in  the  third  row  placed 
with  the  second,  and  then  all  of  these  with  the  first  row. 

In  symbols :  (a  +  h) -\- c  =  a-{-  (h  -\- c). 

This  fact  is  called  the  associative  law  of  addition. 

396.  Graphical  Representation.  The  associative  law  may  be 
represented  graphically  thus : 

a  +  6    +  c 

A 

a  (  h  2 \ 


(  -  t 


a    +    6 


The  properties  stated  above  are  often  used  to  abridge  calculations. 
Thus,  7  +  4  +  3  +  6,  are  more  easily  added  thus :  (7  +  3)  +  (4  +  6). 

ORAL  EXERCISES 

Add  in  the  easiest  way : 

1.  8  +  3  +  2  +  7.  8.  48a;  +  73a;  +  2a;  +  7a;. 

2.  91  +  43  +  9.  9.  192/  +  542/  +  62/  +  2/. 

3.  87  +  26  +  13.  10.  736  +  1866  +  146. 

4.  13  a  +  5a  + 17  a  +  5a.         11.  279^  +  347^  +  21^. 

5.  7  iB  + 12  a; +  3  a; +  18  37.  12.  624p +  45p  +  6p  +  5p. 

6.  8  2/ +  10  2/ +  7  2/ +  5  2/.  13.  93^  +  9^  +  7  ^  +  f. 

7.  23a  +  6a  +  2a  +  4a.  14.  144m  +  7m  +  6m  +  3m. 

WRITTEN   EXERCISES 

Show  graphically  that : 

1.  11 +  4 -I- 6  =  11 +  (4 +  6).       3.    8  +  5  =  5+8. 

2.  3+(4  +  l)=l  +  3  +  4.  4.  4a  +  56  =  56  +  4a. 


294  A  HIGH   SCHOOL  ALGEBRA 

397.  Subtraction.  It  often  happens  that  we  wish  to  know 
how  many  objects  are  left  when  some  of  a  set  are  taken  away, 
or  to  know  how  much  greater  one  number  is  than  another. 
The  process  of  finding  this  number  is  called  subtraction.  The 
number  taken  away  is  called  the  subtrahend,  that  from  which 
it  is  taken,  the  minuend,  and  the  result,  the  difference  or  the 
remainder. 

398.  The  sign  of  subtraction  is  — . 

399.  Subtraction  is  the  reverse  of  addition,  and  from  every 
sum  one  or  more  differences  can  at  once  be  read. 


Thus,  froi 

n  5  +  7 

=  12  we  read  at  once  12  - 

5 

=  7, 

12- 

7 

=  6. 

And  from  5  -f-  5  = 

=  10,  we  read  10  -  5  =  5. 

And  from 

a  +  5  z 

=  c,  we  read  c  —  a  =  b, 
c  —  b  =  a. 

Likewise, 

from  a 

-\-  h  -\-  c  =  d  we  read  d  —  a 

= 

&  +  c, 

d-(a  +  b) 

= 

c,  etc. 

400.  There  is  no  commutative  law  of  subtraction.  For  7—4 
is  not  the  same  as  4  —  7.  In  fact,  the  latter  indicated  differ- 
ence has  no  meaning  in  arithmetic.  We  cannot  take  a  larger 
number  of  objects  from  a  smaller  number. 

401.  In  algebra,  where  numbers  are  often  represented  by 
letters,  we  may  not  know  whether  the  minuend  is  larger  than 
the  subtrahend  or  not.  For  example,  in  a~b,  we  do  not 
know  whether  a  is  larger  than  b  or  not.  But  it  is  desirable 
that  such  expressions  should  have  a  meaning  in  all  cases,  and 
this  is  accomplished  by  the  definition  and  use  of  relative 
numbers. 

402.  The  First  Extension  of  the  Number  System.  ,  Relative 
Numbers.  Whenever  quantities  may  be  measured  in  one  of 
two  opposite  senses  such  that  a  unit  in  one  sense  offsets  a  unit 
in  the  other  sense,  it  is  customary  to  call  one  of  the  senses  the 
positive  sense,  and  the  other  the  negative  sense,  and  numbers 
measuring  changes  in  these  senses  are  called  positive  and 
negative  numbers  respectively.    (For  examples,  see  Chapter  IV.) 


REVIEW  AND   EXTENSION  OF   PROCESSES        295 

403.  A  number  to  be  added  is  offset  by  an  equal  number  to  be 
subtracted ;  hence  such  numbers  satisfy  the  above  definition, 
and  numbers  to  be  added  are  called  positive,  and  those  to  be 
subtracted  are  called  negative.  Consequently,  positive  and 
negative  numbers  are  denoted  by  the  signs  -f  and  — 
respectively. 

Thus,  4-  5  means  positive  five,  and  denotes  five  units  to  be  added  or  to 
be  taken  in  the  positive  sense. 

—  5  means  negative  five,  and  denotes  five  units  to  be  subtracted,  or  to 
be  taken  in  the  negative  sense. 

404.  Graphical  Representation.  Relative  integers  may  be 
represented  graphically  thus : 

-5    -4   -3   -2    -I       0    +1    +2   +3  +4  +5 

It  appears  that  the  positive  integers  are  represented  by  just  the  same 
set  of  points  as  the  natural  or  absolute  integers.  For  this  and  other 
reasons  the  absolute  numbers  are  usually  identified  with  positive  numbers. 
Although  it  is  usually  convenient  to  "do  this,  we  have  in  fact  the  three 
classes  of  numbers  :  the  absolute,  the  positive,  and  the  negative.  Thus, 
we  may  consider  $  5  without  reference  to  its  relation  to  an  account,  or  we 
can  consider  it  as  $  5  of  assets,  or  we  may  consider  it  as  $  5  of  debts. 

405.  According  to  Sec.  403,  the  signs  +,  —  denote  the  oper- 
ations of  addition  or  subtraction,  or  the  positive  or  negative 
character  of  the  numbers  which  these  signs  precede. 

If  it  is  necessary  to  distinguish  a  sign  of  character  from  a 
sign  of  operation,  the  former  is  put  iyito  a  parenthesis  with  the 
number  it  affects. 

Thus,  +  8  —  (—  3),  means  :  positive  8  minus  negative  3. 

WJien  no  sign  of  character  is  expressed,  the  sign  plus  is  un- 
derstood. 

Thus,  5  —  3  means  :  positive  5  minus  positive  3. 
Similarly,  Sa  +  9a  means  :  positive  8 a  plus  positive  9 a. 

406.  Absolute  Value.  The  value  of  a  relative  number  apart 
from  its  sign  is  called  its  absolute  value. 

20  ,.. ,. 


296  A  HIGH  SCHOOL  ALGEBRA 

ORAL    EXERCISES 

Kead  the  following  in  full,  according  to  Sec.  405  : 

1.  6-4.  9.  12-(-5).  17.  7-9. 

2.  -5-8.  10.   -12-(+5).  18.  7  +  9. 

3.  -8  +  20.  11.   -7 -(-9).  19.  -7 +(-9). 

4.  2a  +  3a.  12.  2a-(4-3a).  20.  2y-(-Sy). 

5.  26-36.  13.  c-\-d.  21.  -2x-(-3x). 

6.  —2a- 3a.  14.  c— d  22.  -2x-{-(—3x). 

7.  2?/+(+32/).  15.  m+(-n).  23.  -26-(-5c). 

8.  3p_(-2p).  16.  4:X+{-2x).  24.  3a-(-\-5y), 

WRITTEN    EXERCISES 

Indicate,  using  the  signs  +,  —  : 

1.  The  sum  of  positive  5  and  positive  3. 

2.  The  sum  of  positive  a  and  negative  6. 

3.  The  difference  of  positive  p  and  positive  q. 

4.  The  difference  of  negative  5  and  positive  3. 

5.  The  difference  of  negative  x  and  positive  y. 

6.  The  sum  of  positive  a  and  positive  6. 

7.  The  sum  of  negative  a6  and  negative  a6. 

8.  The  sum  of  positive  y  and  negative  x. 

9.  The  difference  of  positive  xy  and  negative  xy. 
10.   The  difference  of  negative  pq  and  positive  mn. 

407.   Addition  of  Relative  Numbers. 

Just  as  3  pounds  +  5  pounds  =  8  pounds, 

so  3  positive  units  +  5  positive  units  =  8  positive  units, 

and         3  negative  units  +  5  negative  units  =  8  negative  units. 

To  add  units  of  opposite  character,  use  is  made  of  the  de- 
fining property  of  relative  numbers,  that  a  unit  in  one  sense 
offsets  a  unit  in  the  other  sense.  Thus,  to  add  3  positive  units 
and  7  negative  units  we  notice  that  the  3  positive  units  offset 


REVIEW  AND   EXTENSION  OF  PROCESSES        29T 

3  of  the  negative  units  and  the  result  of  adding  the  two  will 
be  4  negative  units. 

That  is,  (  +  3)  +  (  -  7)  =  (  +  3)  +  (  -  3)  +  (  -  4)  =  -  4. 

In  general : 

I.  If  two  relative  numbers  have  the  same  sign,  the  absolute 
value  of  the  sum  is  the  sum  of  the  absolute  values  of  the  addends, 
and  the  sign  of  the  sum  is  the  comm^on  sign  of  the  addends. 

II.  If  two  relative  numbers  have  opposite  signs,  the  absolute 
value  of  the  sum  is  the  difference  of  the  absolute  values  of  the 
adde7ids,  and  the  sign  of  the  sum  is  the  sign  of  the  addend  having 
the  larger  absolute  value. 

408.  More  than  two  numbers  are  added  by  repetition  of  the 
process  just  described.     This  may  be  done  either  : 

(1)  by  adding  the  second  number  to  the  first ;  then  the  third 
number  to  the  result,  and  so  on;  or 

(2)  by  adding  separately  all  the  positive  numbers  and  all  the 
negative  numbers,  and  then  adding  these  two  results. 

409.  It  may  be  verified  that  the  Commutative  and  the  Asso- 
ciative Laws  of  Addition  hold  also  for  relative  integers. 

410.  Subtraction  of  Relative  Nximbers.  Since  n  units  of  one 
sense  are  offset  by  adding  n  units  of  the  opposite  sense,  we 
may  subtract  n  units  of  one  sense  by  adding  n  units  of  the 
opposite  sense. 

Thus,  7_(+3)=7+(-3). 

And,  7_(_3)=7+(+3). 

And,  4-(  +  7)  =  4+(-7). 

411.  Accordingly,  subtraction  may  be  regarded  as  the  inverse 
of  addition :   To  subtract  a  monomial,  we  add  its  opposite. 

To  subtract  an  algebraic  expression  consisting  of  more  than 
one  term,  we  subtract  the  terms  one  after  another. 

In  general,  to  subtract  any  algebraic  expression  we  may  change 
the  sign  of  each  of  its  terms  and  add  the  result  to  the  minuend. 


298  A  HIGH   SCHOOL  ALGEBRA 

ORAL  EXERCISES 

State  the  sums : 

1.  5  +  (-3).  4.    -12z  +  (-lSz). 

2.  -6a  +  (-7a).  5.   11  a;-|- ( -2a;)  +  (- 5  a;). 

3.  -lly  +  Sy.  6.    -3g  +  7g  +  (-6g). 

7.  How  may  the  correctness  of  a  result  in  subtraction  be 
tested  ?     State  the  differences : 

8.  11-6.         10.    -lla-(-6a).     12.    -31y-(-3y). 

9.  -11-6.     11.  31a; -(4-5  a;).         13.    Up -(-Up). 

14.  How  may  a  parenthesis  preceded  by  the  sign  +  be  re- 
moved without  changing  the  value  of  the  expression?  One 
preceded  by  the  sign  —  ? 

15.  How  may  terms  be  introduced  in  a  parenthesis  preceded 
by  the  sign  -f-  without  changing  the  value  of  the  expression  ? 
In  a  parenthesis  preceded  by  the  sign  —  ? 

WRITTEN    EXERCISES 


Add: 

1.   2a  +  5 

6.    c-\-d  —  5 

11.   4:X  —  2z  +  y 

a  +  4 

c—  d-j-5 

2x—     y  +  z 

2.  3a  +  8  7.    x  +  y  +  2z  12.    l+rn^-^p"^ 

a  —  4  x  —  y-\-4:Z  1  —  m^  —  p"^ 

3.  6  6+     c  8.  p  +  g—     m  IZ.   ax-\-hy  ^cz"^ 
3  6  —  2  c                    p—  q-\-2m  bx-^ay—  z^ 


7. 

x-\-y-{-2z 
x  —  y-\-4:Z 

8. 

p  +  q—    m 
p  —  q  -\-2m 

9. 

2x-    2/+    2 
2x+2y-A.z 

0. 

ax-\-by-\-  c 
ax-{-   y  -^  c 

4.    -3a-{-  6          9.   2x-    y-\-  z      14.  1.5x+3.5y-\-    z 

2a-Sb  2x-\-2y-4.z  .5x-{-6.5y-.lz 

5..  4a  — 5  10.   ax-\-by-\-c  15.   i^  +  |.V  — i^; 

3a  +  7  aa;H-2/  —  c  ^ 

Subtract : 

16.   4a  +  6  17.        8a;H-3  18.   llx-4:y 

2a-9  -3x^2  19a; 


REVIEW   AND  EXTENSION  OF  PROCESSES        299 


19. 

12 1 

22. 

16y  + 

—  z 

25.    -Ilm4-40i) 

-6t-\-3 

82/- 

10  z 

-40m-12j9 

20. 

7x  +  Si/ 

23. 

10X2_ 

■16  2/2 

26.       x  —  7y-\-5z 

2y-4.x 

20x2-f 

■   42/2 

—  x-{-4:y-6z 

21. 

4a  +  36 

24. 

41a;2- 

16  2/2 

27.     p-\-     q—     m 

2c-5a 

15a^2_ 

20  2/2 

Qp  —  2q-\-4m 

28. 

ax"^  +  52/2  +  c 

31. 

2.5 

a  +  6.36-.lc 

ax"^  —  by^  —  c 

1.5 

a-3.55-.9c 

29. 

m2  +  2j)2_ 

6g2 

32. 

ix 

-|2/  +  i^ 

—  m^  —  4:  p^ -\- 

5g2 

ix 

-i2/  +  t^ 

30. 

40  x'  - 10  2/2  +    z'^ 

33. 

a^x 

_|_  &2^  _^  ^2^2 

50x2  +  40/- 

-7  22 

a^x  H-  by^  +  C2!2 

Remove  parentheses  and  unite  terms  as  much  as  possible : 

34.  3a-2  5+(3  6-7a). 

35.  4m  +  5-(63-3p). 

36.  {llx  +  5y)-{-Zx-\-2z). 

37.  7 -[2 -(3 -5)]. 

38.  (4a4-2a)-[(7a-5a)  +  (-6a-17a)]. 

39.  3  +  \^x--2-(l  x-l-2-{-3x)\. 
^0.  4x+\2x  —  \3-(l  x^^)  —  l-\-x']\. 

41.  a6-J2a6  +  c-[3c-(6-a5)]j. 

42.  12xy-\2xy  —  Q>z^[4.xy—{2z-\rxy)']\. 

43.  -{:Spq-3xy  +  l)-\-2pq^Zxy-{xy  +  2)\. 

44.  —  ;  4  a6c  -  (2  ac  +  6c)  j  +  f  6  ahc  +  (2  ac  -  &c)  | . 

Write  the  expressions  of  Exercises  45-56,  as  x  minus  a  paren- 
thesis. Also  group  the  terms  involving  x  and  y  in  each  expres- 
sion in  a  parenthesis  preceded  by  the  sign  — . 

45.  3a  +  i»  +  2  2/.  48.    2mH-p  +  aj  — 2/. 

46.  —5 2/  + 7c  — 8  + a;.  49.    a-\-x—b  +  cy. 

47.  Q>t'^  +  4y-{-x  —  3.  lyO.  p  +  x  —  y-\-by. 


300  A   HIGH   SCHOOL  ALGEBRA 

51.  x  —  3b-\-2y  —  c.  5^.   2p  —  q-\-x  —  y. 

52.  Sb-[-x  +  ay-\-by.  55.    ay  +  x  —  z  -\-  zy. 

5Z.    by  -^  m  —  n  -\-  X.  56.    c  +  (^  +  a;  —  (a  +  b)y. 

412.  Multiplication.  To  multiply  two  absolute  integers 
means  to  use  the  one  (called  the  multiplicand)  as  addend  as 
many  times  as  there  are  units  in  the  other  (called  the 
multiplier).     The  result  is  called  the  product. 

Thus,  3  times  4  means  4  4-4-1-4. 

The  simpler  products  are  obtained  by  actually  making  the  additions  that 
are  implied.  For  large  numbers,  as  we  have  seen  in  arithmetic,  the  pro- 
cess may  be  much  abridged  by  use  of  properties  of  the  decimal  notation. 

413.  Commutative  Law  of  Multiplication.  The  expression  3 
times  5  means  5  -f-  5  -}-  5,  and  may  be  indicated  as  follows  : 

0  9  «  #  «  That  is,  since  each  horizonal  line  (or  row) 
contains  5  dots,  there  are  all  together  3  times 
5  dots.     But  each  vertical  line  (or  column) 

•     •     •     •     •    contains  3  dots  and  there  are  5  columns. 

Hence  there  are  5  times  3  dots  all  together.     But  the  number  of 

dots  is  the  same  whether  we  ^ 

count  them  by  rows  or  by  col-  r — - — - — — -^ — n 

umns,  hence  5  times  3  equals 

S  times  5.     Quite  similarly,  if 

we  have  a  rows  of  dots  with  b 

dots  in  each  row,  it  follows  that 

a  times  b  equals  b  times  a. 


t     t     f     •     t- 


"^     4     4     4     4 ^ 

The  result  may  be  stated  in  symbols  thus : 

ab  =  ba. 

This  fact,  called  the  Commutative  Law  of  Multiplication, 
means  that  the  product  is  not  altered  if  multiplier  and  multi- 
plicand are  interchanged.  Consequently,  these  names  are  fre- 
quently replaced  by  the  name  factor  applied  to  each  of  the 
numbers  multiplied. 


REVIEW   AND   EXTENSION  OF  PROCESSES        301 

414.  Let  each  dot  of  the  block  of  dots  of  (A)  above  have  the 
vahie  of  6.  Since  there  are  3x5  dots,  the  value  of  the  block 
would  be  (3  X  5)  X  6. 

A  second  expression  for  the  value  of  the  block  is  obtained 
by  finding  the  value  of  the  first  row  (viz.  5x6)  and  multiply- 
ing it  by  the  number  of  rows,  or  3.  The  expression  resulting 
must  be  equal  to  that  already  found,  or : 

3  x  (5  X  6)  =  (3  X  5)  x  6. 

Similarly,  if  each  dot  of  the  block  (B)  above  has  the  value 
c,  it  follows  that 

a  (be)  =  (ab)c 

That  is  :  The  product  of  three  absolute  integers  is  not  altered 
if  they  be  grouped  for  multi plication  in  any  way  possible  without 
changing  the  order.  This  is  a  case  of  what  is  known  as  the 
Associative  Law  of  Multiplication  for  absolute  integers. 

415.  By  similar  methods  and  use  of  these  results  it  can  be 
proved  that  both  the  Commutative  Law  and  the  Associative 
Law  apply  to  all  products  of  absolute  integers.     That  is  : 

Commutative  Law.  The  product  of  any  number  of  given  factors 
is  not  changed,  if  the  order  of  the  factors  be  changed  in  any  way. 

Associative  Law.  The  product  of  any  number  of  given  factors  in 
a  given  order  is  not  changed  if  the  factors  be  grouped  in  any  way. 

The  Distributive  Law.     From  the  block  of  dots  we  see  that 


This  is  called  the  Distributive  Law  of  Multiplication,  and  the 
above  proof  covers  the  case  in  which  a,  b,  and  c  are  absolute 
integers. 


302  A   HIGH   SCHOOL  ALGEBRA 

416.  Multiplication  of  Relative  Integers.  To  multiply  by  a 
positive  integer  means  to  take  the  multiplicand  as  addend  as 
many  times  as  there  are  units  in  the  multiplier,  and  to  multi- 
ply by  a  negative  integer  means  to  take  the  multiplicand  as 
subtrahend  as  many  times  as  there  are  units  in  the  multiplier. 

Consequently, 

4.3  =  3  +  3+3  +  3  =  12. 
4(-3)=-3  +  (-3)  +  (-3)  +  (-3)=-12. 
(_4)3=_3_3_3_3=-12. 
(_4)(-3)  =  -(-3)-(-3)-(-3)-(-3)  =  12. 

So,  generally, 

(+a)(+6)  =  +a6, 
(+a)(-6)  =  -a6, 
(-a)(+5)  =  -a6, 
(— a)  (-?>)= +a6. 

417.  In  words:  The  product  of  two  {integral)  factors  of  like 
signs  is  positive,  and  of  two  factors  of  unlike  signs  is  negative; 
in  each  case  the  absolute  value  of  the  product  is  the  product  of  the 
absolute  values  of  the  factor. 

418.  We  observe  that  the  Commutative  Law  holds  in  this 

case  also.  » 

» 

For  example  :  (—b)a  =  a(—b).  Since  ba  =ab,  by  the  Commutative 
Law  for  absolute  integers,  and  by  Sec.  416,  (— b)a  = —ba  = —ab  = 
a(-6). 

It  may  be  shown  that  the  Associative  and  the  Distril^fflve 
Laws  also  hold. 

ORAL    EXERCISES 

State  the  laws  that  are  applied  in  the  various  steps  of  the 
following  calculations : 

1.  2. 8. 3. 5  =  2. 5. 8. 3=  10.  24  =  240. 

2.  5(17  .  2  - 6  c)  =  5(17  •  2)  -5(6  c)  =  {5'  2)17  -  (5  .  6)c 

=  10  .  17  -  30  c  =  170  -  30  c. 

3.  7b(5x  +  ab)  =  (7  b)  (5  x)  +  (7  b)  (ab)  =  35  bx  +  7  ab\ 


REVIEW   AND   EXTENSION   OF  PROCESSES        303 

4.     {a  +  b){G-{-d)  =  a{c-{-d)  +  b{c-{-d)  =  ac-{-  ad  +  5c  +  bd. 


State  the  products : 
5.    ax 
5b 

8. 

a'b 

a^c 

11.        6  xyz 

-2x^ 

6.    -2a 
-3a2 

9. 

5x' 
2%^ 

12.    -4a2^ 
-3&2^ 

7.         8  a; 
-4.xy 

LO. 

-x^y 
-xy"^ 

13.    -5afy 
5a^y 

14.  {a-rby. 

15.  {a-cy. 

16.  {x-^yy. 

17.  (x-ay. 

18.  (a;-2  2/)^ 

19.  (x-\-y)(x- 

20.  (2  a; -1)2. 

21.  (2m  +  n)2. 

■y)' 

/'■ 

22. 
23. 
24. 
25. 
26. 
27. 
28. 
29. 

(a;-l)(a.  +  l). 

(1-2/)  (1+2/). 
(m  +  2a;)2. 

(3-2d)(3  +  2(^). 
(4-0^)1 
-3^x^-2). 
(l-2d){l+2d). 

a^'-i2/)a^+i2/). 

%' 

TTEN    EXERCISES 

Multiply  and  test : 
*1.   2a  +  3 

2a  +  4 

3. 

a;  +  2 
a;-3 

5.   /-32/  +  4 
y-2 

2.   3  a -2b 
2a-Sb 

4. 

x-2 

a;  +  l 

6.   _p-3^  +  ^ 

J9-2i 

7.  (Zx-4tyy.  14.  (aP-6"»-l)2. 

8.  (52/-70(82/  +  20.  15.  {4.x-3xy){2x  +  2  f). 

9.  (6a  +  13g)2.  16.  {ax-^by){cx^dy). 

10.  (l  +  a.')(2-ha;)(l-a;).  17.  (ax" -by)(cx-dy'). 

11.  (1  — 2  2/)l  18.  (m  +  ^+p)(w«'  +  ^— i>). 

12.  (6-3)(6  +  7)(6-3).  19.  (a  +  6  +  c-d)l 

13.  (a;"  +  2/")(aj~-r)-  20.  {a-2b){a -2  h)(a-2b). 


304  A  HIGH  SCHOOL  ALGEBRA 

419.  Detached  Coefficients.  —  When  two  polynomials  can 
be  arranged  in  descending  powers  of  the  same  letter,  the  work 
of  multiplication  may  often  be  shortened  by  using  the  coefficients 
only.     This  is  called  multiplication  by  detached  coefficients. 

Thus,  to  multiply  3x2  —  2x+lbya;— 1: 

Work  in  Full  Wokk  with  Coefficients  Only        Test 

3x2-2x  +  l  3-2  +  1  x  =  12 

x-l 1-1  0 

'     3x3-2x2+     X 

-3x2  +  2x-l 


3x3-5x2  +  3x-  1 


3-2  +  1 

x  =  l 

1-1 

3-2+1 

-3+2-1 

3_5+3_l 

The  processes  in  the  two  cases  are  identical  with  the  exception  of  the 
omission  of  the  letters  in  the  second  case.  The  product  of  3  x2  and  x,  the 
first  terms  of  the  factors,  show  that  the  first  term  of  the  product  is  3  x^, 
the  next  term  must  contain  x2  and  the  next  x,  because  the  terms  in  the 
result  must  be  in  order  of  degree.  Thus  we  may  perform  the  operation 
with  coefficients,  and  then  supply  the  proper  letters  and  exponents. 

When  some  powers  of  the  letters  are  missing,  zeros  must  be 
supplied  as  coefficients  of  the  missing  powers  in  order  to  keep 
a  record  of  the  places  in  which  the  pow^s  are  missing. 


Thus,  to  multiply  2x3— 7x  +  3by2x 
2  x8  -7  X  +  3  as  2x-^  +  0  x2  -  7  X  +  3, 
and    perform    the   multiplication    a; 
shown  at  the  right. 


X  —  5,  write 

2  +    jl^  7  +  3 
2-    5 

■15 

4  + 

0-14  +  6 
10-    0  +  35- 

4- 

10x3 

10  _  14  +  41  - 
-  14  x2  +  41  X 

15 
-  15 

Test 
-2 


+  6 


.-.  (2x-5)(2x3-7x  +  3)  =4x4 


WRITTEN   EXERCISES 

Multiply  by  detached  coefficients  and  test : 

1.  w  +  n                 3.    a2  +  2a+-l                 5.  f-y^+5 
2  m  — Sn                g  +  l  2y+-3 

2.  af-5x  +  S       4.   a;4_3a^+.5                6.  x* -^  2  x"^ -\- 7  x 
x-l                       x-\-2  2x-l 


7.  x'^-x-6hj2x^-4:X-16. 

8.  a2_^_i2by  3a2-15a+-18. 


REVIEW  AND  EXTENSION   OF  PROCESSES        305 

9.   a4-2a2  +  lby  a*  +  2a2  +  l. 
10.  m^  —  m^  —  m  +  1  by  2  m  —  1. 

12.  a;2  -f  a??/  H-  y^  by  x^  +y^  —  xy. 

13.  a;2»  — 4x"  4-4  by  1  — ic'^+ic^'*  — 2£c2«. 

420.  Division.  Division  is  the  process  of  finding  a  number 
called  the  quotient,  such  that  when  multiplied  by  a  given  num- 
ber, called  the  divisor,  the  product  is  a  given  number,  called 
the  dividend. 

421.  The  fundamental  relation  of  division  is  : 

Divisor  times  Quotient  equals  Dividend. 

From  this  it  follows  at  once  that  if  dividend  and  divisor 
have  the  same  sign,  the  quotient  is  positive,  and  if  they  have 
unlike  signs,  the  quotient  is  negative. 

422.  In  dividing  polynomials,  it  is  best  to  arrange  both  divi- 
dend and  divisor  in  the  order  of  the  powers  of  the  same  letter. 


Thus, 

(.r3  _  3  x2  +  3  a;  -  1)  --  (x  -  1). 

instead  of 

(3x-3ic2  +  a;3-l)  -^  (x-1). 

WRITTEN    EXERCISES 
Divide : 

1.  Qi? -^x'^  —  x—lhj  x-\-l.         4.   ic3«  —  2/^«  by  03"  —  2/^ 

2.  x^  —  1  by  x  —  1.  5.   a;^  —  1  by  x^  —  1. 

3.  y^ —  z^hy  y'^  +  yz  +  z^.  6.  p^ -\- q^  \^J  p -{- q- 

7.  m^  -f  m^  +  m  -f  1  by  m  +  1. 

8.  x^  —  px  —  qx-\-pq\)YX—p. 

9.  4cc4+l  by  2aj2._2aj+l. 

10.  x^  —  2xy  —  ^y'^hj  X  —  3  y. 

11.  6  a^  -  10  a2  + 13  a  _  6  by  3  a  -  2. 

12.  6  7?i^  -  2  m^  -  8  ?7i2  +  4  m  -  8  by  3  7^2  _  ^  _j_  2. 

13.  a?^"  -f-  ic«  _  6  by  ic«  —  2. 

14.  x^  -^4.  x^^y  -f  6  aj2«2/2  -f  4  x^f  -j-  y*  by  x"^"  +  2x''y  +  yl 


306  A  HIGH   SCHOOL  ALGEBRA 

423.  Detached  Coefficients.  When  both  divisor  and  divi- 
dend involve  a  regular  series  of  powers  of  the  same  letters,  it 
is  easier  to  divide  with  coefficients  only. 

Thus,  to  divide  oi^  —  ^x^-\-^x—\\)yx  —  l. 


Work  in  Full 

Work  with  Coefficients  Only 

x2  -  3  X  +  1 

1-3  +  1  or  x2-3x  +  l 

X  -  \)xi  _4a:^  +  4x-l 

1_  1)1-4  +  4-1 

x^-x'^ 

1-1 

-3x2 

-3 

-  3  x2  +  3  X 

-3  +  3 

x-1 

1-1 

x-1 

1-1 

Test.    Letting 

X 

=  2, 

(2  -  1)  (4  -  6  +  1)  =  8  -  16  +  8  -  1. 
.  -1=-1. 

424.   If  the  series  of  powers  is  not  complete,  zero  coefficients 
must  be  used. 

Thus,  x3  +  3  x2  +  1  must  be  regarded  as  x^  +  3  x^  +  0  x  +  1,  and  the 
coeflQcients  1  +  3  +  0  +  1  must  be  used  in  the  division. 

WRITTEN    EXERCISES 

Divide  by  detached  coefficients  and  test : 

1     ^!zil.  2        ^'-1  3     ^^+1. 

'    a  —  1  'a2  +  a  +  l  '2a;4-l 

4.  (a;2  +  5a;+6)-=-(aj  +  2). 

5.  (a;2-2a;-3)--(a;-3). 

6.  (10  +  a;2_7aj)^(a;-2). 

8.    (a^+-3a32+-3a;+-l)--(a;  +  l). 

10.  (6«  +  H-9a;2)^(3aj  +  l). 

11.  (a^  +  a2  4- 1)  ^  (a2  _  a  +  1). 
j2    m^  +-  4  m^  +  6  m^  +-  4  m  +- 1 

m2  -f  2  m  + 1 


REVIEW  AND  EXTENSION  OF   PROCESSES        307 

425.  Special  Quotients.  The  general  form  of  quotients 
represented  by  the  type  (ic"  ±  ?/")  -i-(x±y)  may  readily  be  seen 
by  division. 

Thus,  considering  (a;**  +  ?/**)  -f-  (a:  +  y),  we  have 


By  inspection,  what  do  you  think  the  fourth  term  of  the  quotient  will 
be?    The  fifth  ?    Verify  your  opinion  by  continuing  the  above  division. 


WRITTEN    EXERCISES 

Find  the  quotient  and  remainder  if  any  : 

1.  (a:5_l)^(a;_l).      5.    {x'-y^)^{x-^y). 

2.  (o^^V)^(x-l).      6.    {^-y^)^(x-y). 

3.  (a^  +  l)-^(.T  +  l).       7.    (a^  +  2/')^(^  +  2/)- 

^^     x'-y^  ^2.    ^!±l!.         13^    x'-y\         ^^ 

x—y  '    x—y  '    x+y 


9. 
10. 


a;7  _  y-r 

x  —  y 

x'^f 

x-y 

x^  +  jf 

x  +  y 

15.  Substitute  5  for  w  in  the  first  five  terms  of  the  above 
division,  and  compare  with  the  result  of  Exercise  7. 

16.  Substitute  6  for  n  similarly  in  the  first  six  terms,  and 
compare  with  the  result  of  Exercise  14. 

17.  Use  w  =  7  in  the  first  7  terms ;  compare  with  Exercise  8. 

18.  Take  n  =  5  and  y  =  1,  and  compare  with  Exercise  3. 

19.  Divide  x""  +  y"  by  x  —  y  to  seven  terms.  Let  7i  =  6  and 
compare  with  the  result  in  Exercise  12.  Let  n  =  7  and  com- 
pare with  the  result  in  Exercise  10. 

20.  Divide  x""  -y""  by  x  +  y  to  six  terms.  Let  ri  =  6  and 
compare  with  the  result  of  Exercise  13. 

21.  Divide  x'^  —  y''  by  x  —  y  to  six  terms.  Let  ?i  =  6  and 
compare  with  the  result  of  Exercise  11. 


308  A  HIGH   SCHOOL  ALGEBRA 

ZERO  AND  ITS  RELATION  TO  THE  PROCESSES 

426.  Definition  of  Zero.  Zero  may  be  defined  as  the  result 
of  subtracting  a  number  from  itself. 

a-a  =  0. 

427.  Addition.  By  definition  of  zero,  a  +  0  =  a+6— 6  =  a, 
since  to  add  b  and  immediately  take  it  away  leaves  the  original 
number  a. 

a  +  0  =  a. 

428.  Subtraction.  Similarly,  a  —  0  =  a  —(b  —  b)  =  a  —  b  -\- 
b  =  a,  since  to  take  away  b  and  then  replace  it  leaves  the 
original  number  a. 

a-0  =  a. 

To  add  or  subtract  zero  does  not  alter  the  original  number. 

429.  Multiplication.     By  definition  of  zero, 

0  •  a  =  (6  —  b)a  =  ba  —  ba  =  0. 
Or,  0  •  fl  =  0. 

That  is,  if  one  factor  is  zero,  the  product  is  zero. 

Multiplication  by  zero  simply  causes  the  multiplicand  to  vanish. 

It  follows  directly  that  -  =  0. 

430.  Division.  According  to  the  definition  of  division  a  -^  0, 
or  - ,  asks :  By  what  must  zero  be  multiplied  to  produce  a  ? 

Let  X  denote  the  desired  number.     Then  0  '  x  =  a. 
But  we  know  that  zero  times  any  number  is  zero.     If  a  is 
not  zero,  there  is  no  number  x  that  satisfies  the  above  equation. 

That  is,  ?  =  no  number. 
0 

If  a  is  zero,  every  number  x  satisfies  the  equation. 
That  is,  -  =  any  number,  since  0  times  any  number  =  0. 

Division  by  zero  is  therefore  either  entirely  indefinite  or  im- 
possible.    In  either  case  it  is  not  admissible. 


REVIEW  AND   EXTENSION  OF   PROCESSES        309 

If  we  divide  one  literal  expression  by  another,  there  is  no 
guarantee  that  the  result  is  correct  for  those  values  of  the 
letters  that  make  the  divisor  zero. 

EXAMPLE 

Let  a  =  6.  {1) 

Multiplying  both  members  by  a,  a^  =z  ab.  {2) 

Subtracting  &2  from  both  members,  Op'  —  h'^  =z  ab  —  b^  (3) 

Factoring,  (a  +  b)  {a  —  b)  =  b(a  -  b).  (4) 

Dividing  both  members  hy  a-b,  a  -\-  b  =  b.  (5) 

Substituting  the  value  of  a  from  (1),  b  +  b  =  b.  (6) 

Or,  2b  =  b.  (7) 

Dividing  by  &,  2  =  1.  (8) 

The  work  is  quite  correct  to  equation  (4) .  But  by  dividing  equation  (4) 
by  an  expression  that,  according  to  the  conditions  of  the  problem,  is  zero, 
we  find  as  the  result  an  incorrect  equation. 

FRACTIONS 

431.  The  Second  Extension  of  the  Number  System;  Frac- 
tions. We  found  the  primitive  or  absolute  integers  by  counting. 
We  defined  the  operation  of  addition  for  these  integers  and 
saw  that  it  was  always  possible.  Next  we  defined  the  opera- 
tion of  subtraction  for  these  integers,  and  found  that  it  was 
not  always  possible.  This  led  us  to  define  relative  numbers 
(positive  and  negative).  In  the  system  of  numbers  as  enlarged 
by  this  first  extension,  we  saw  that  both  addition  and  subtrac- 
tion are  always  possible.  Then  we  defined  multiplication  for 
all  integers  and  saw  that  it  was  always  possible.  We  now 
examine  the  operation  of  division  as  just  defined. 

The  operation  12  -^  4  is  possible  in  the  system  of  integers 
because  there  exists  an  integer,  3,  whose  product  with  4  is  12. 
But  the  operation  12  -f-  5  is  impossible  in  the  system  of  integers, 
since  there  exists  no  integer  whose  product  with  5  is  12. 
This  leads  us  to  define  another  kind  of  number,  the  fraction. 
This  is  done  by  dividing  the  unit  into  b  equal  parts,  and  taking 

a  of  these  parts.     A  symbol  for  the  new  number  is  -,  in  which 

a  and  b  may*  be  any  integers.     This  constitutes   our  second 
extension  of  the  number  system. 


310  A  HIGH   SCHOOL  ALGEBRA 


432.  The  new  number,  -,  is  called  a  fraction ;  a  is  called  the 

numerator  and  h  the   denominator  of  the  fraction;  a  and  h 
together  are  called  the  terms  of  the  fraction. 

433.  Graphical  Representation  of  Fractions.     Fractions  may 

a   ..  be  represented  by  points  of 

jT 31 M  I  1  I  I  I  I  ^  ■  I     the   number   scale  that  we 

'      ^     '       T  have  already  had. 

Thus,  if  the  distance  from  zero  to  1  is  divided  into  6  equal  parts,  and 
then  a  of  these  parts  laid  off  from  zero^  the  end  point  represents  the  frac- 


tion 


-•     (In  the  figure  the  fraction  is  -.  )     Fractions  may  also  be  taken 
h       \  5  / 

in  the  negative  sense.     Two  fractions  are  said  to  be  equal,  or  to  have  the 

same  value,  when  they  are  represented  by  the  same  point  of  the  number 

scale. 

434.  If  each  of  the  h  equal  parts  is  halved,  making  2  h  parts, 
and  each  of  the  a  parts  taken  is  halved,  making  2  a  parts,  the 
end  point  remains  the  same. 

That  is : 

g  _2a 

h~2h' 
Similarly : 

a  _na 

b      nb 

(A)  In  words :  The  value  of  a  fraction  is  not  altered  if  both 
numerator  and  denominator  is  multiplied  by  the  same  integer. 

(B)  Reading  the  above  equation  from  right  to  left:  TJie 
value  of  a  fraction  is  not  altered  if  both  numerator  and  denomi- 
nator be  divided  by  a  common  integral  factor. 

Every  integer  may  be  regarded  as  a  fraction. 

Thus,  3  =  ^  or  ^  or  ^  etc.     a  +  6=^+-^. 
12         5  1 

435.  Addition  and  Subtraction  of  Fractions.  The  properties 
(A)  and  (B),  Sec.  434,  enable  us  to  add  and  subtract  fractions. 

If  fractions  have  not  the  same  denominator,  they  are  first  reduced  to 
the  same  denominator,  and  then  the  results  added. 


REVIEW  AND  EXTENSION  OF   PROCESSES        311 

WRITTEN    EXERCISES 

Add  and  express  the  results  in  lowest  terms : 


510  a  5         4                      6_l6  +  l 

^    .T  — 1        a;  +  4    ^  ^    _2 3^ 

'    x-\-l      2(a;+l)'  ■    a+p      (a+pf 

5    ^  ,  3.y  5      _3 ^ 

'    2y     4.x  '   x'     20^ 

a  —  bb  1  —q'      1  +q 

y 


^°-    (1  _  »)(1  _  2,)  +  (2/  -  x){y  _  1)  +  (,,  _  1)(1  _  2/) 
n.     „     ,.^     ..    +    .     „.^ +  1 


12. 


2xy  -\-y^  _     y     _  of -\- 5  xy 
{x-yf      x  —  y      (x  +  yy 


436.  The  addition  and  reduction  of  fractions  is  facilitated  by 
the  processes  for  finding  the  1.  c.  m.  and  the  h.  c.  f .  The  methods 
for  finding  these  by  factoring  given  in  Chapter  XIV  are  suffi- 
cient for  elementary  algebra. 

WRITTEN  EXERCISES 
Find  the  I.e.  m.  of: 

1.  48,60,120.  4.    x^-x,  x^-2x-{-l. 

2.  15a2,  21a%  105  abxy.      5.   a^-5x-6,  x^-j-6x-]-5. 

3.  5  abx,  25  a%  4:0  b'x^  6.   a{x-l),  b{l-x''),  c(x-a^), 

7.  {l-3xy,  l-dx",  l-4a;  +  3a;l 

8.  m^  +  m  +  1,  m^  —  m  + 1,  m^  -\-im?-\- 1. 

9.  Sar'-lScc  +  O,  4.x'-llx-3. 

10.  a;-l,  (^-1)3,  .T2-2a;  +  l. 

11.  (29-g)Q9H-r),  (q-r){q-p),  (r  -  q)(r -\- p). 

12.  (a  —  c){a  —  b),  (c  —  a){b  —  c),  {b  —  a){c  —  b). 

13.  a^-6x''  +  llx-6,x'-4:X-\-3,x^-3x-^2. 

21 


312  A   HIGH   SCHOOL   ALGEBRA 

Findtheh.c.f.  of: 

14.  28,  35,  105.  16.   l-a,a'-2a-{- 1. 

15.  21  x",  42  ax',  63  abx^y.         17.   1- x\  1- x%  a^  -  2x-i-l. 

18.  a^  +  2  ab  +  b%  a^  +  ?>^ 

19.  x-\-y,  x^  —  xy',  x^  —  2xy  —  3y^. 

20.  a;'*  +  x'y^  +  y^,  a^  +  ?/^  —  a:?/. 

21.  a3  +  5a2-6  a,  2a2-2. 

22.  3z^-7z'--h4r,5^-17z^  +  16z-4.. 

23.  a^  +  62  +  2  c2  +  2  a6  +  3  ac  +  3  6c,  ac  +  6c  +  cl 

437.  Fractions  may  sometimes  be  added  more  easily  by 
adding  them  in  an  order  different  from  that  in  which  they 
were  given. 

EXAMPLES 

1.    In  adding 1 f-  -^^ — | it  is  especially 

^a^la-3a+la  +  3  ^  "^ 

easy  to  add  the  first  and  third  fractions ;  then  the  second  and 

fourth ;  and,  finally,  the  results  thus  obtained. 

Thus, 

1  1         a+l-a+1  2 


a  —  1      a  +  1  a^  —  1  a^  —  1 

1       ,       1     _a+S+a-S        2a 


a-S     a +  3  a2-9  a'^  -  9 

2  2  a    ^2  a^  -  IS  -\-  2  a^  -  2  a  ^2(a^  -\-  a^  -a  -  9)  ^ 

2.   In  adding -A^  +  -4t  +  -^  +  4!^4   ^1^^   ^^^^   ^^^ 
a—b      a-jrb      a^  -\-b^      cc  +  b^ 

second  are  easily  combined,  then  that  result  is  easily  added  to 

the  third,  and  finally  that  result  to  the  fourth. 

,^.  3      ,      3  fJa 

(2) 
(3) 


a -b     a+b 

cfi  -  b-' 

Qa      .      Qa     _ 
a2  _  &2     a2  +  62 

12  a3 

a'^-b^ 

12^3         12a3    _ 

a4  _  54       «4  +  ?,4 

24  a7 

a8_68 

REVIEW  AND   EXTENSION  OF   PROCESSES        313 

WRITTEN     EXERCISES 

1.  Add  the  fractions  in  the  examples  above  in  the  ordinary 
way ;  that  is,  find  the  1.  c.  d.  of  the  several  fractions  and  add 
at  once. 

Add  by  successive  combinations,  as  above : 

2.   ^-+     1  2  a 


a  —  X     a-\-  X     a^  +  cc^ 
3.    ^+      ■'  3 


a  — 1      a  — 2      a-\-2     a  +  1 

2      ,      2       .      4a 
4.    r  + 


a  —  b      a  -^b      a^  +  6^ 

1     _iCi  —  b         1         a-{-b 


a—  b         c  a  -\-b         c 


i>4-l     p-2     p  +  2     p-\-l 

7.    Add  several  of  the  preceding  problems  in  the  ordinary 
way.     Which  method  is  shorter  ? 

438.  To  multiply  a  fraction  by  an  integer  we  extend  the 
definition  of  multiplication  to  this  case,  and  we  have : 

3(1)=! +  *  +  *  =  -¥- 

Similarly,    (-  c)  Z'^')  =  -  ^  -  -  ••  •  to  (c  terms)  =  -  — • 

439.  That  is,  to  midtiply  a  fraction  by  an  integer  we  multiply 
the  numerator  by  that  integer. 

The  fraction  -  is  the  quotient  called  for  in  the  indicated  division  a  -^  6, 

because  the  product  of  the  divisor  b  and  the  asserted  quotient  ^  is  the 
dividend  a,  as  seen,  in  ^ 

I).  ^  =  ^=a    (by  property  J5,  Sec.  434). 
b       b 

In  our  present  number  system  the  fraction  -  may  therefore  always  be 

b 
regarded  as  indicating  the  division  of  a  by  6  (provided  b  is  not  zero) . 


314  A   HIGH   SCHOOL  ALGEBRA 

440.  The  operation  of  division  is  thus  seen  to  be  possible  in 
the  new  number  system  for  any  integral  dividend  and  any 
integral  divisor  (except  zero). 

We  shall  see,  under  Division  of  Fractions,  that  if  either  dividiend  or 
divisor  or  both  are  fractions,  the  operation  is  still  possible  without  en* 
larging  the  number  system  further. 

441.  Multiplication  of  Fractions.  The  product  of  two  or 
more  fractions  is  a  fraction  whose  numerator  is  the  product  of  the 
given  numerators  and  whose  denominator  is  the  product  of  the 
given  denominators. 

442.  Since  an  integer  or  an  integral  expression  may  be  re- 
garded as  a  fraction  with  denominator  1,  this  definition  applies 
also  when  one  or  more  of  the  factors  are  integral. 

443.  The  Associative,  Commutative,  and  Distributive  Laws 
of  Multiplication  apply  to  fractions  as  well  as  to  integers. 

WRITTEN    EXERCISES 


Multiply : 

1.     ^   .^^ 
la   15  b 

5. 

c2    2  b' 

2    0?    ^xy^ 

'     y2      4^5 

6. 

-3a    46 
a;2-9    9  a; 

3.     ''-^    .^'-^. 
il  +  qf   {2^ty 

7. 

(HJ- 

Sy    Wa^y   15t^x 

8. 

<^-ll 

0.  ,qr(^-ff. 

^^       a{l-a)            1  + 

a 

l  +  2a-fa2   1-2  a 

i  +  a^ 

11         ^-^'               ^- 

ib-\-a 

<1- 

b     \ 

•   i^2b  +  b^    62_2f 

-a     1  —  bJ 

12. 


t~l  t-\-l  1  —  ty  4:t 


REVIEW  AND  EXTENSION  OF   PROCESSES        315 

444.  Reciprocals.  If  the  product  of  two  numbers  is  1,  each 
is  called  the  reciprocal  of  the  other. 

Thus,  the  reciprocal  of  -  is  -,  since  ^  .  ^  z=  ^  =  1. 
ha  h    a     ah 

445.  Division  of  Fractions.     Since  divisor  x  quotient  =  divi- 

a         c 
dend,  to  divide  -  by  -  means  to  find  a  number  q  such  that 
0         d 

d    ^      b 

Solving  the  above  equation,  we  have, 

^      b      c 

That  is :  To  obtain  the  quotient,  multiply  the  dividend  by  the 
reciprocal  of  the  divisor. 

We  have  thus  seen  that  in  the  number  system  as  now  extended  the 
four  fundamental  operations  are  possible  when  any  or  all  of  the  numbers 
involved  are  fractions.     (Division  by  zero  is  always  excepted.) 

446.  Complex  Fractions.  If  one  or  both  of  the  terms  of  a 
fraction  are  themselves  fractions,  the  given  fraction  is  called 
a  complex  fraction.  Since  a  fraction  indicates  division,  a 
complex  fraction  may  be  simplified  by  performing  the  indi- 
cated division  of  its  numerator  by  its  denominator. 


WRITTEN    EXERCISES 

Divide,  and  express  results  in  lowest  terms : 


1. 

X  .  y 
2  '  6' 

4.   ^«%2 
46 

c\ 

7  ^^' .  ^g\ 

'    2    ■    8 

2. 

ab    .4  62 
18  c  '  3  ac 

5.    t-U 

i  h 

0^  +  1 

2  m 

S.  pv^  +  ^-^. 

3 

a«2  ^  bx^ 
2f  '  ay'^ 

6.     -^^- 
4y  +  l 

.      1 

-6  .  -18 

o. 

*  2a;-l 

10. 

5x 

'/■      n. 

2x 

1 

x 
3* 

12.  .f  "• 

3  a^ 

4:X 

13  ^•^^ 

1 
14.  p.       15. 

9m3 

X 

26 

316  A  HIGH  SCHOOL  ALGEBRA 


16.    {t±^-a]-.a-^ 


17.    ^  +  ^^fl„^ 
b      d     \        bd 


^        o?)     \a2     of 


Perform  the  operations  indicated : 
19.    ^ L_.    20.    t+^.    21.    to  +  lV(l-^- 


1+-  7+1 

X  t 


FACTORING 

447.  Factoring.  The  process  of  finding  two  or  more  factors 
whose  product  is  a  given  algebraic  expression  is  called  factoring 
the  given  expression. 

In  division  one  factor  (the  divisor)  is  given  and  another  factor  (the 
quotient)  is  to  be  found  such  that  the  product  of  tlie  two  factors  is  the 
given  expression  (the  dividend).  Division  is  thus  really  a  variety  of 
factoring,  although  the  name  "factoring"  is  not  usually  applied  to  it. 
As  a  rule,  when  an  expression  is  to  be  factored,  none  of  the  factors  is 
specified  in  advance,  and  any  set  of  factors  is  acceptable  on  the  sole  con- 
dition that  their  product  is  the  given  expression, 

448.  The  type  products  of  special  importance  were  applied 
in  Chapter  XII  and  are  collected  here  for  reference. 

I.    xy  -j-xz  =  x  (y  -f  z). 
IL    x-'±2xy+y''={x±yy. 

III.  x^  —  y^  =  (x  ^-  y)  {x  —  ?/) . 

IV.  x'^-\-{a-\-b)x-\-ab  =  (x-^a)(x-\-b). 

V.  acx^  +  (ad  -{- be)  x -{- bd  =  (ax  +  b){cx  -j-  d). 

VI.  x^±3x''y-\-3xy^±f  =  (x±  yf: 

VII.  x^  -\-y^  =  (x  +  y)  {x^  —  xy-{-  y"^).     . 

VIII.  0?  —  y^  =  {x  —  y) (x^  -\-xy-^y^). 

These  formulas  apply  when  the  letters  are  either  positive  or  negative 
numbers,  and  a  detailed  treatment  of  them  was  given  in  Chapter  XL 
The  following  miscellaneous  exercises  cover  all  the  types. 


REVIEW  AND  EXTENSION  OF  PROCESSES        317 


ORAL    EXERCISES 


Factor : 

1.  ax -{-ay. 

2.  ax  -\-  a. 

3.  ax-\-a'^. 

4.  abx  —  ay. 

5.  x'^  -\-  ax. 

6.  a^  +  ax^. 

7.  aV  —  ax. 

8.  mx  H-  my  +  m. 

9.  jpaj^+p^ic+pg. 

10.  a^  —x'^-\-x. 

11.  a^y  +  a??/ -h  2/- 

12.  (a  +  6)a;  —  (a  +  %. 

13.  sH-{-st^+  sH^' 

14.  igt'-\-gt. 

15.  abc  —  acd -\- bed. 

16.  a^x^-bh/. 

17.  4£c2— 9:^2^ 

18.  a262a;2_c2. 

19.  l-p^qY' 

20.  m^p'^-sH\ 

21.  (a  +  &y-(c  +  d)2. 

22.  l-(m+i))2. 

23.  a2  4.2a6c  +  63c2^ 

24.  aV  +  2a2a;2  +  l. 

25.  rt2_^2ci(6-|-c)  +  (6  +  c)' 


26.  l_2(a;4-2/)  +  (a'  +  ^)' 

27.  49  4-14a;  +  aj2. 

28.  9  -  12  ?/2  H- 4  2/^ 

29.  16-40  2+2522. 

30.  aW-{-2abcd-{-cW. 

31.  1  +  ^  +  ^'. 

32.  2/^ -h  .4  2/2  +  .04, 

33.  m^iK^  +  4  mcc  +  4. 

34.  ^V^4_i^2  +  i. 

35.  a;2— 5a5  +  6. 

36.  a;2^7^_^i2. 

37.  x'^-x—12. 

38.  a;2+a;-6. 

39.  i2_i2a;  +  35. 

40.  s2-2a;-35. 

41.  12a;2  +  7a;+l, 

42.  15  2/2 -2  2/ -1. 

43.  6a;2-12a;  +  6. 

44.  32/2  +  82/  +  5. 

45.  a^-bK 

46.  a3  +  6^ 

47.  8  -  a^d^. 

48.  64:-a^f. 

49.  8a^-12a^2_|_ga;-l. 

50.  27^3  +  272/2+92/  +  l. 


WRITTEN    EXERCISES 


Factor : 

1.  x'^-49. 

2.  t''~7t-\-6. 


3.  5f-Sy^-{-y. 

4.  4v2_9^2^ 


318  A  HIGH  SCHOOL  ALGEBKA 

5.  l4-10a  +  25a2.  26.  .125a^ -{- .75x^-^1.5 x  +  l. 

6.  a3  +  3a26  +  3a62  +  6l  ^^  ^_^.  §^_1 

7.  (x-^iy-{.2(x  +  l)  +  l.  *  8        4         2 
l_?_Li  ^^-  8a3  +  12a2^4-6a«2  4.^^ 

tt     a2  29.  ^^-16. 

9.  x^-2x-15.  30.  ay -11 012/ +  24. 

10.  aW-1.  31.  4/i2-12/ii  +  9«2, 

11.  a;4  +  6ic22^  +  92/2.  32.  1  _  m^. 

12.  'y'-^'.  33.  (2a  +  by-(3a-2by. 

13.  l-a^2/'.  34.  8a3-32a^ 

14.  a^-4a262.  ^^  10  m^a;  -  60  ma;  +  90  a;. 

15.  (x  +  2/)^-l.  3g^  my  +  4s^ 

16.  a363_c3d3.  ^^  a;^  +  3a;^-28. 

17.  .^-2.y  +  ,i  33   ^3_e4. 

18.  (m+pY  —  l. 

)         d     ,  „  39.   56  a;2- 68  a; +  20. 

19.  (x  +  yy-(p  +  qy. 

on     )   Tm3_l^  40.   24x2  +  a;-10. 

21.  a;3+|^2 4.3^+1.  41.  15a;^-34a;-16. 

22.  a^-lia^  +  fa-i..  42.  ah' -  7  az -{- 12. 

23.  a;2  +  (a  +  6)a;  +  a6.  43.  2/^-132/2  +  42. 

24.  y'-{-(ac-bd)y-abcd.  44.  (a  + 1)^  -  (a  -  1)2. 

25.  acx"^ -{:  (cb -{- ad)x -\- bd.  45.  aj^  — ia;  — ^. 

Calculate : 

46.    272-252.  47.   3872-3772.  48.   263-253. 

General  Methods 

449.  The  General  Trinomial.  The  factors  of  the  general 
trinomial  mx"^ -\- px -{- q  have  the  form  (ax-\-b)(cx-{-d),  and 
they  may  be  found  by  reducing  the  general  form  to  the  type 
a;2  +  6a;  +  c.  To  do  this,  multiply  and  divide  the  expression 
by  m  (the  coefficient  of  x"^)  and  put  mx  =  y.  The  method  will 
be  made  sufdciently  clear  by  an  example. 


REVIEW  AND  EXTENSION  OF   PROCESSES        319 

EXAMPLE 
Factor:  6a;2  + 19  a;  + 10. 

Multiplying  and  dividing  the  given  trinomial  by  6,  we  obtain 

6x-2  +  19x  +  10  =  K36a;2  +  6  •  19  x  +  60). 

Put                                       Qx  =  y.  (i) 

Then,         36  x2  +  6  .  19  a;  +  60  =  2/2  +  19  y  +  60  {2) 

=  (2/  +  4)(2/  +  15)     Sec.  163.  (5) 

Replacing  y  by  6a?,                               =(6  X  +  4)  (6  iC  +  15)  (4) 

=  2(3x  +  2)3(2x+5) 

=  6(3a;  +  2)(2x  +  5).  (5) 
And  finally,          6  x2  +  19  X  +  10  =  ^  •  6(3  iC  +  2)(2  X  +  6) 

=  (3x  +  2)(2x  +  5).  {6) 

Test  by  multiplication. 


WRITTEN    EXERCISES 
Factor : 

1.  3^2 -h  7  a;  4- 2.  17.  7  p^  4.  22  p  +  3. 

2.  3«2_5a;-2.  18.  ISa^  +  ^-G. 

3.  4a;2  +  13a;-12.  19.  ^a''  +  Uah-Bh\ 

4.  6a;2-22a;  +  20.  20.  15 a;2  +  2 cc?/ -  8 2/^. 

5.  3a2  +  a-10.  21.  14  m^  -  m^;  -  3  pi 

6.  4m2  +  9m-9.  22.  10^2+98^-9^2^ 

7.  4p2-|_8j9  +  3.  23.  12  a;4  -  47  a;2  -  17. 

8.  9a;2_3x-2.  24.  40  i»2a  _  2  a^a  _  2. 

9.  5x2_9^_2.  25.  3a--8a-115. 

10.  6a2_8a-8.  26.  Q  0?^"  -  aW  -  12  h\ 

11.  25  2/2 -50  2/ -24.  27.  25a^  +  20a;  +  4. 

12.  10a;2  +  9a;2/-92/l  28.  4:  z^ -\- 4:  az  -  15  a\ 

13.  15^)2  _i8p- 24.  29.  3a'  +  ab-2b\ 

14.  12  s2- 10  s -12.  30.  12  2/2 +  19  2/ -21. 

15.  62/2  +  352/  — 6.  31.  6m^-{-mn  —  2n\ 

16.  28;32_44-g_,.16.  32.  96  a^- 4  a6  -  15  5^ 


820  A   HIGH   SCHOOL   ALGEBRA 

450.  Expressions  that  can  be  made  the  Difference  of  Two 
Squares. 

1.  Expressions  of  the  form  a'^"'  +  4  b"^""  may  be  factored  by 
adding  and  subtracting  4  a^^b^"".     Thus, 

=  (a2"  +  2  52-  _  2  a"6«)  (a^-  +  ^^n  +  2  a"6'^) . 

EXAMPLE 

a:8  _|_  4  y8  _  aj8  _^  4  ^.4^4  +  4  ^s  _  4  ^^ 

=  (X*  +  2  2/*  -  2  x2?/2)  (a4  +  2  y4  +  2  x2?/2). 

2.  Expressions  of  the  form  a'''*  ±  j^a^^ft^^  _|_  547.  j^g^y  j^g  factored 
if  a  number  can  be  found  such  that  when  added  to  p  the  result 
is  2. 

EXAMPLE 

ractora4-6a262  +  64^ 

Adding  and  subtracting  4  a^b^, 

a*  -  6  a262  +  &*  =  a*  -  2  a^b^  +  &*  -  4  a252, 
=  (a2  _  62)2  _  4  ^252. 

=  {a^-b'^-2abXa^-b^  +  2ab). 

3.  Some  other  expressions  may  be  made  the  difference  of 
two  squares. 

EXAMPLE 

Factor  16  m*  +  36  mY  +  ^5  p\ 

Since  twice  the  product  of  the  square  roots  of  the  end  terms  is  40  m'^p^, 
the  expression  can  be  made  the  difference  of  two  squares  by  adding  and 
subtracting  4  m2p2.    Thus, 

16  m*  +  36  m2p2  +  25 p^  =  16  m*  +  40  m^p"^  +  26p^-4  m^p'^ 
=  (4m2  +  5p2)2_(2mp)2 
=  (4  m2  +  5  ;?2  _  2  mp)  {4:m^  +  5p^  +  2  mp). 

WRITTEN    EXERCISES 

Factor : 

1.  y  +  494.  4.   4?/8  +  L  7.    a^  +  a^  +  l. 

2.  m^  +  4.  5.   2/^2 _^4,  8.    a4  +  4a2  +  16. 

3.  4a;4^1.  6.    4:m^-\-y^.  9.    a^  +  a262 -f  54^ 


REVIEW  AND   EXTENSION   OF   PROCESSES        321 

10.    64p*-fl.  11.    m»  +  64.  12.    4:p^-\-q^. 

13.  9a;4-10a;2+l.  15.    Ac'-^-S  cW +  9  d\ 

14.  16  ^4 +  4 2/2 -hi.  16.    36x^-S4.xy  +  16y\ 

451.  The  General  Binomial.  The  factors  of  the  general  bi- 
nomial a;"  ±  y""  depend  upon  the  nature  of  n.  The  following 
table   shows   for  what  values  of  n  the  division  is  exact  in 

a.  (x''-\-  y"")  -^  {x  +  y),  if  n  is  odd. 

h.  {x^  +  2/")  -5-(ic  —  y),  not  for  any  value  of  n. 

c.  (i«"  —  y^)  -^  (ic  -h  2/),  if  n  is  even. 

c?.  (ic''  —  2/")  -i-  (ic  —  2/),  for  every  integral  value  of  n. 

ORAL  EXERCISES 
Name  a  factor  of  each  of  the  following : 

1.  x^  —  y^.  3.    1-x^.  5.    a^io  — 1.  7.    a^^  — 6". 

2.  ic^  +  al  4.    aj»  +  al  6.    x^^-y^\         8.   32aj5-l. 

452.  The  factor  of  the  first  degree  in  x'^  ±  y'^  having  been 
found  by  inspection,  the  other  is  a  regular  series  of  powers 
that  can  be  written  directly  (Sec.  425) ;  it  has  alternate  signs 
in  the  cases  (a)  and  (c),  and  plus  signs  in  case  (d). 

For  example : 

x^  ^a'  ^{x^  a) (xP  -  ax^  +  a'^x^  -  a^^  +  a%2  _  gb^  +  ««). 

32a5-l=(2a)5_  1 

=  (2  a  -  1) [  (2  a)4  +  (2  a)3  .  1  +  (2  a)2  .  12  +  2  a  .  13  +  1*] 

=  (2a-l)(16a*  +  8a3  +  4a2  +  2a  +  l). 

a;io  _  7.10  has  both  x  —  r  and  a;  +  r  as  factors,  but  it  is  better  first  to 
factor  as  a  difference  of  two  squares ;  tlien  apply  (d)  and  (a). 

WRITTEN    EXERCISES 
Factor : 

1.  32a^-65.  5.  a' -b\  9.  {x-^yf^l. 

2.  128  a^  +  1.  6.  1  -  m^o.  10.  ^  -  2/^  (four  factors). 

3.  32- 243  «^  7.  x'' -  a\  11.  {a-\-hY -{a-h)\ 

4.  (c-d)7  +  l.  8.  l-{a  +  h)\  12.  (a;_2/)4-(x  +  2/)^ 


322  A  HIGH   SCHOOL   ALGEBRA 

453.  Factor  Theorem.  If  any  polynomial  in  x  assumes  the 
value  zero  when  a  is  substituted  for  x,  then  x  —  sl  is  a  factor  of  the 
polynomial. 

For  example,  substitute  2  for  x  in  the  polynomial : 

Then,  2*  -3  •  2^  +  7  •  22  -  2  •  2  -  16  =  16  -  24  +  28  -  4  -  16  =  0. 

Suppose  the  polynomial  to  be  divided  by  oj  —  2,  and  denote  the  quo- 
tient by  Q  and  the  remainder  by  B,  the  latter  being  numerical. 

Then,  x^-Sx^+7x^-2x^l6={x-2)Q+  B. 

In  this  equation  substitute  2  for  ic ;  the  left  number  becomes  zero  as 
just  seen  ;  2  —  2  is  also  0,  and  0  times  §  is  0  ;  hence  the  result  is  : 

0  =  0  +  J?,  or  i?  =  0. 

Consequently,  7^ —  ^x^  -\-1  x^ —  2x—lQ  —{x —  2)Q,  or  x  —  2  is  a 
factor  of  ic*  —  3  a;3  4-  7  a;2  —  2  a:  —  16.  Thus  we  may  know  without  actual 
division  that  x  —  2  is  a  factor  of  the  polynomial. 

In  general :  Let  P(x)  denote  the  given  polynomial. 

Suppose  P(x)  to  be  divided  by  a;  —  a.  There  will  be  a  certain  quotient, 
call  it  Q{x),  and  a  remainder,  B.  This  remainder  will  not  involve  x, 
otherwise  the  division  could  be  continued. 

We  have  then:         P(x)  =  (x  -  a)Q(x)  +  B.  (1) 

Put  a  for  X,  and  denote  the  resulting  value  of  P(x)  by  P(a),  and  of 
Q{x)  by  Q{a) : 

Then,  P(a)  =  (a- a)(Q)(a)  +  B.  (2) 

But,  (a-a)(Q)a  =  0, 

hence,  if  P(a)  =  0, 

then,  B  =  0, 

and,     ,  P(x)  =  (x  — a)Q{x). 

That  is,  cc—  a  is  a  factor  of  P(x),  if  the  expression  is  0  when  x  =  a. 

WRITTEN     EXERCISES 

By  use  of  the  Factor  Theorem  test  each  expression  and  find 
if  the  binomial  at  the  right  is  a  factor  of  it : 

Expression  Binomial 

1.  x^—4.x-\-4:.  x  —  2. 

2.  a;2— oa;-f6.  x  —  S. 


REVIEW  AND  EXTENSION  OF  PROCESSES       323 

3.  x'^-x-6.  x-3. 

4.  Sx'^  —  x  —  2.  x  —  1. 

5.  3  2/2 -20?/ +  25.  '                    y-5. 

6.  a^-3a;24-3a;-l.  x-1. 

7.  jp^  — 4p2— 4j9+ 16.  j?  — 4. 

8.  m^  —  2  m^  +  m  —  2.  m  —  5. 

454.   If  the  polynomial  becomes  0  when  —  a  is  put  for  x,  then 
X  —  (—  a),  or  X  +  a  IS  a  factor. 

Example  :  x  +  2  is  a  factor  of  o;^  +  a;^  _  2  a;  because 

(-  2)3  +  (-  2)2  _  2(  -  2)  =-  8  +  4  +  4  =  0. 

WRITTEN     EXERCISES 

By  use  of  the  Factor  Theorem,  prove  that  each  polynomial 
has  the  factor  named : 


Polynomial 

Factor 

1. 

aj3  +  12  a;2  ^  31  a;  -  20. 

x-{-5. 

2. 

{x  -  ay  -{-(x-  by  -  (a  -  by. 

X-b. 

3. 

s^^2x^-^Sx-{-2. 

x-\-l. 

4. 

0-      2^2+^^     (m^      2m2+^^V 
m      V                        a;  y 

X  —  m. 

5. 

aj3»  +  2  a;^"  +  3  aj"  4-  2. 

a;"  +  l. 

6. 

a^ -j- aa^  -  aV  —  al 

X—  a^. 

7. 

9  aj5  +  (3  a2  _  12  a)  a;4  -  4  a^aj^  +  3  a^a;  +  a*. 

-f 

In  each  polynomial,  substitute  the  values  given  for  x,  and 
use  the  Factor  Theorem  to  find  the  possible  factors :      - 

Polynomial  Values  for  te 

8.  a;4  +  3  a^  +  4  a;2  -  12  X  -  32.  1,  2,  -  2. 

9.  x*-7x^-\-Ux''-^x-  21.  1,  -  1,  2,  3. 

10.  2  aj4  +  5a^  -  41  a;2  -  64  a;  +  80.  4,  -  4,  5,  -  5. 

11.  Apply  the  Factor  Theorem  to  (a"  ±  5")  -5-  (a  ±  &)  and  prove 
the  results  of  Sees.  451  and  452. 


324  A   HIGH  SCHOOL  ALGEBRA 

REVIEW 

WRITTEN    EXERCISES 

Perform  the  indicated  operations,  expressing  fractional  re- 
sults in  lowest  terras : 

1.  2a-Sb-\-4:a  +  llc-2d-\-6-Sa-9b  +  Sc-4:-5d 

-\-2a-b. 

2.  oify-\-Sxy^-\-4.x^  —  2xy  —  8o[^y-\-2  ^y^  —  4:Xy-\-Sxy^ 

-Ty'-Sxy. 

3.  5x-7 y-{3  x -{- 4.y- 2)+ 3 -\-(Sx- 7) -(2 x-Sy-hlS) 

+  8(2  0^-1). 

4.  4  a;- J  2  c  — (3  0^  +  2  2/  + 5  c)|. 

5.  5ac-j46  +  2c[3ci-56-(6a-26-7c-[-3)]J. 

IiL  =  4.x-{-2y-S,   M=x-9y-\-l,   72  =  ?/- 5a;,  find: 

6.  L-^M+B.  8.    LM-3R.  10.    {L  +  Mf. 

7.  3X-2il^+i?.       9.   L''-M\  11.   3U-6MR. 

IfX=7a  +  25,    r=2a-9&  +  3,   Z=4  6- 7  a- 3,  find: 
12.    X-{3Y-2Z).  13.    2F-[Z+3(4X-8  F)]. 

14.    X2-16ri  15.    Y^^-A:YZ.  16.    XFZ. 

17    oHh_5_aM:_75     a  — 5 

a-^     0^-2^     a  +  6 

6m  +  4p     Sm^-\-4:p^     m  —  2p 
15  m  15  mp  5p 

19.    l_9+ii_^.  20.    a-iVx  +  2/). 

l  +  a«      a  +  1  V«     2/7 

21.    „       2  1  3 


^2^9^_^20      ^2_^7^  +  12     ^2^11^  +  28 

24.  (6 +  3)3 -(6 -2)3. 

25.  (a;**  +  3  a;^"^  +  5  a;''-^)  (a;^  _^  4  ^j). 
5  a;3      5  a;^\      15  a; 


/lO^ 

V3m2 


26.    .  „      ^     , 

9  m^ 


REVIEW   AND   EXTENSION   OF   PROCESSES        325 


28. 


Ib  +  ^g  +  g'    '  24.  +  llg  +  g'' 

1_3^  g  +  1  J  g-1 

29.    LjL'.  30.    ^-^      ^  +  ^. 

2^_3  g  +  1      g-1 

a^     a  q  —  lq-\-l 

Set     llb\f3a       b\     fS  a      2  5^^ 

6        3a  jl  5  3aj     [  b       3a 


31. 


32.      .       '         „+  1  ^ 


33. 


34. 


cc'^//     ^    2  .'«^ 

4  g  4-  7         .  16  ft-  +  56  ft  +  49 
a2_l6aj  +  64  *  a^  -  64 


Divide  by  detached  coefficients  : 

35.  16-32x-\-24.x'-Sx^-\-x*---(2-x). 

36.  16m^-32m'^  +  2477i2_8m  +  l^(2m-l). 
Factor  by  the  Factor  Theorem : 

37.  x^-i-x'^-x  —  1. 

38.  4aj4  +  a^  — 5. 

39.  a^  —  ab^  +  a^  —  ab.     (Substitute  b  for  a.) 

SUMMARY 

1.  State  the  laws  that  govern  the  processes  of  addition  for 
algebraic  numbers.  Sees.  389,  395. 

2.  State  the  laws  that  govern  the  processes  of  multiplication 
for  algebraic  numbers.  Sees.  412,  418. 

3.  State  and  illustrate  two  extensions  of  the  number  sys- 
tem of  algebra.  Sees.  402,  431. 

4.  Name  eight  type  expressions  used  in  factoring.     Sec.  448. 

5.  State  the  Factor  Theorem.  Sec.  453. 


326 


A   HIGH   SCHOOL   ALGEBRA 


HISTORICAL  NOTE 

We  have  seen  that  algebra  embraces  the  enlargement  of  the  number 
field  of  arithmetic  to  include  negative  number  and  the  extension  of  the 
basic  processes  to  those  numbers.  Thus,  the  equation  a:  +  3  =  1,  which 
Ahmes  could  not  solve  and  which  was  regarded  as  negligible  until  the  six- 
teenth century,  is  no  more  exceptional  than  x  +  1  =  3,  since  algebra  de- 
fines 1  —  3  to  be  —  2.  In  a  similar  way  the  field  of  number  was  enlarged 
to  include  positive  and  negative  fractions,  for  the  equation  3  x  =  —  2  could 


In  arithmetic  the  processes 


2 

not  be  solved  until  was  understood. 

h  —  a  and  -  are  limited  to  positive  numbers,  but  in  algebra  each  of  these 
a 

has  a  meaning  for  all  values  of  h  and  a,  positive  or  negative.  This  process 
of  generalizing  was  first  explained  by  the  English  algebraist,  George  Pea- 
cock (1830),  and  called  by  him  the  principle  of  permanence.  That  is,  in 
order  that  any  number  or  symbol  may  be  made  a  part  of  algebra,  it  must 
conform  to  the  basic  laws  governing  the  processes,  namely.  The  Associa- 
tive Law,  The  Commutative  Law,  and  The  Distributive  Law. 

The  famous  scientist.  Sir  Wil- 
liam Rowan  Hamilton  (1840), 
regarded  these  laws  as  distin- 
guishing algebraic  number  from 
other  numbers.  In  doing  so,  he 
discovered  numbers  which  do 
not  obey  the  Commutative  Law 
of  Multiplication,  and  to  these 
numbers  he  gave  the  name 
Quaternions.  Their  study  has 
since  become  a  new  branch  of 
mathematics. 

Hamilton  was  of  Scotch  par- 
entage, but  Ireland  shares  his 
fame,  because  he  was  born  and 
educated  at  Dublin.  Like  Tar- 
taglia,  he  received  instruction  at 
home  when  a  boy,  and  showed 
exceptional  ability  at  an  early 
age.  When  only  thirteen  he 
could  read  a  dozen  languages, 
at  eighteen  he  had  mastered  Newton's  Priiicipia,  and  shortly  became  pro- 
fessor of  Astronomy  in  Trinity  College,  Dublin.  Hamilton  did  much  for 
mechanics  and  astronomy,  but  his  greatest  achievement  in  mathematics 
was  the  discovery  of  Quaternions. 


Sir  William  Rowan  Hamilton 


CHAPTER   XXV 

EQUATIONS 

EQUATIONS  WITH  ONE  UNKNOWN 

455.  Two  algebraic  expressions  are  equal  when  they  repre- 
sent the  same  number. 

456.  If  two  numbers  are  equal,  the  numbers  are  equal  which 
result  from  : 

1.  Adding  the  same  number  to  each. 

2.  Multiplying  each  by  the  same  number. 

Subtraction  and  division  are  here  included  as  varieties  of  addition  and 
multiplication. 

457.  The  equality  of  two  expressions  is  indicated  by  the 
symbol,  =  ,  called  "  the  sign  of  equality." 

458.  Two  equal  expressions  connected  by  the  sign  of  equality 
form  an  equation. 

459.  Such  values  of  the  letters  as  make  two  expressions 
equal  are  said  to  satisfy  the  equation  between  these  expressions. 

460.  Equations  that  are  satisfied  by  any  set  of  values  what- 
soever for  the  letters  involved  are  called  identities. 

461.  Equations  that  are  satisfied  by  particular  values  only 
are  called  conditional  equations,  or,  when  there  is  no  danger  of 
confusion,  simply  equations. 

462.  The  numbers  that  satisfy  an  equation  are  called  the 
roots  of  the  equation. 

463.  To  solve  an  equation  is  to  find  its  roots. 

464.  The  letters  whose  values  are  regarded  as  unknown  are 
called  the  unknowns. 

22  327 


328  A   HIGH   SCHOOL   ALGEBRA 

465.  The  degree  of  an  equation  is  stated  with  respect  to  its 
unknowns.  It  is  the  highest  degree  to  which  the  unknowns 
occur  in  any  term  in  the  equation.  Unless  otherwise  stated, 
all  the  unknowns  are  considered. 

466.  An  equation  of  the  first  degree  is  called  a  linear 
equation. 

467.  An  equation  of  the  second  degree  is  called  a  quadratic 
equation. 

468.  An  equation  of  the  third  or  higher  degree  is  called  a 
higher  equation. 

469.  In  order  to  state  the  degree  of  an  equation  its  terms 
must  be  united  as  much  as  possible. 

470.  Terms  not  involving  the  unknowns  are  called  absolute 
terms. 

471.  Equivalent  Equations.  If  two  equations  have  the  same 
roots,  the  equations  are  said  to  be  equivalent.  If  two  equa- 
tions have  together  the  same  roots  as  a  third  equation,  the  two 
equations  together  are  said  to  be  equivalent  to  the  third. 

472.  The  Linear  Form,  ax  +  b.  Every  polynomial  of  the 
first  degree  can  be  put  into  the  form  ax  -\-  b.  That  is,  by 
rearranging  the  terms  suitably,  it  can  be  written  as  the  product 
of  a;  by  a  number  not  involving  x,  plus  an  absolute  term. 
Hence,  the  form  ax-\-b  is  called  a  general  form  for  all  poly- 
nomials of  the  first  degree  in  x. 

473.  Every  equation  of  the  first  degree  in  one  unknown  can 
be  put  into  the  form  : 

ax  -\-b  =  0. 

Consequently  this  is  called  a  general  equation  of  the  first  degree 
in  one  unknown. 

474.  General  Solution.  Erom  the  equation  aa?  -f  6  =  0,  (1) 
we  have  ax  =  —  b,  (2) 

and  hence,  x  =  ^^^  •  (3) 


EQUATIONS  329 

475. is  the  general  form  of  the  root  of  the  equation  of 

the  first  degree.     There  is  always  one  root,  and  only  one. 

The  advantage  of  a  general  solution  like  this  is  that  it  leads  to  a  for- 
mula which  is  applicable  to  all  equations  of  the  given  form. 

In  words :  When  the  equation  has  been  2^^it  into  the  form 
ax  +  b  =  0  the  root  is  the  negative  of  the  absolute  term  divided 
by  the  coefficient  of  x. 

Test.  The  correctness  of  a  root  is  tested  by  substituting  it 
in  the  original  equation. 

If  substituted  in  any  later  equation,  the  work  leading  to  that  equation 
is  not  covered  by  the  test. 

Results  for  problems  expressed  in  words  should  be  tested 
by  substitution  in  the  conditions  of  the  2^roblem. 

If  tested  by  substitution  in  the  equation  only,  the  correctness  of  the 
solution  is  tested,  but  the  setting  up  of  the  equation  is  not  tested.  Nega- 
tive results  that  may  occur  in  such  problems  are  always  correct  as  solu- 
tions of  the  equations,  but  they  are  admissible  as  results  in  the  concrete 
problem  only  when  the  unknown  quantity  is  such  that  a  unit  of  the  un- 
known quantity  is  offset  by  a  unit  of  its  opposite. 

For  example,  if  the  unknown  measures  distance  forward,  a  negative 
result  means  that  a  corresponding  distance  backward  satisfies  the  con- 
ditions of  the  problem.  But,  if  the  unknown  is  a  number  of  men,  a 
negative  result  is  inadmissible,  since  no  opposite  interpretation  is  possible. 


ORAL    EXERCISES 

Solve  for  x : 

1.  3^7  =  15.  4.  a;- 6  =  10. 

2.  2  a?  =  11.  5.  0^4-6  =  12. 

3.  41  a;  =  9.  6.  2 a; -hi  =  13. 

Solve  f or  ^ : 

7.  6^  =  36.  10.  3^-8  =  22. 

8.  ^  —  5  =  20.  11.  at=:ab. 

9.  2t-\-5  =  2D.  12.  at  +  b  =  c. 


330  A  HIGH   SCHOOL   ALGEBRA 

Solve  for  y : 

13.  3  2/  — 1  =  2.  16.  hy  =  hc. 

14.  2  2/  — 1  =  7.  17.  hy  =  h-\-c. 

15.  5  2/4-5  =  35.  18.  ay  —  b  =  c. 

Solve  for  a : 

19.  6a  =  18.  22.  3a  +  l  =  13. 

20.  21  a  =  10.  23.  2  a -5=  15. 

21.  a  4- 1=13.  24.  5  a- 6  =  c. 


WRITTEN    EXERCISES 
Solve  for  x : 

1.  a; -75  =  136.  11.  325  a;  4- 60  =  400, 

2.  2a;-15  =  45.  12.  125aj  +  5  =  10. 

3.  3a;  +  6  =  48.  13.  8  a;  +  625  =  105. 

4.  5  ic  — 10  =  55.  14.  ax-}-bx  =  c. 

5.  1.5  a;  — 5  =  70.  15.  abx -\- ax  =  ab. 

6.  3.5  a;  — 5  =  100.  16.  cx  +  dx  =  c-\-d. 

7.  2.1  a;  —  41  =  400.  17.  ma;  +  pa;=p  +  g. 

8.  1.3  a;  +  .1  =  1.79.  18.  lx-\-mx  =  l  —  m. 

9.  .25  a; -h -50  =  3.25.  19.  ax -{-bx  =  2(a -{-b). 
10.    .11  a;  +  .11  =  1.32.  20.  2cx-^dx  =  l. 


Solve  and  test : 

21.    2(a;-l)+3(2a;4-5)  =  0. 

22.    ^(^■^^^  +  3  =  2a;  +  l. 
5 

24.    3?/  +  l^l-32,_ 
42,-2     4-4y- 

-  "-i^+V'^^-f^=- 

25.   l+3p_17-5p 

EQUATIONS  331 


Solve  for  c : 
26.   5c  +  3a=ac.  28.   v  =  ct^^-^ 


27. 


4c-l      2c4-5 


2 


3  m  6  m  29.    (a  +  c)(a  —  c)  =  —  (c  -|-  a)^. 

30.   _3^_ 0:^^(0-2X20-7}^     •  3^^    ^_^, 

0-3     0-4        c2-7cH-12  1  +  c^ 

32.  An  inheritance  of  $  2000  is  to  be  divided  between  two 
heirs,  A  and  B,  so  that  B  receives  $  100  less  than  twice  what 
A  receives.     How  much  does  each  receive  ? 

33.  Twenty  steps,  each  of  a  given  height,  are  required  to 
build  a  certain  staircase.  If  each  step  is  made  2  inches  higher, 
16  steps  are  required.    Find  the  vertical  height  of  the  staircase. 

34.  In  a  certain  hotel  the  large  dining  room  seats  three 
times  as  many  persons  as  the  small  dining  room.  When  the 
large  dining  room  is  f  full  and  the  small  dining  room  ^  full, 
there  are  100  persons  in  both  together.  How  many  does  each 
room  seat  ? 

35.  Tickets  of  admission  to  a  certain  lecture  are  sold  at  two 
prices,  one  25  cents  more  than  the  other.  When  100  tickets 
at  the  lower  price  and  60  at  the  higher  price  are  sold,  the  total 
receipts  are  $  95.     Find  the  two  prices. 

36.  Originally,  -LQ-  of  the  area  of  Alabama  was  forest  land. 
\  of  this  land  has  been  cleared,  and  now  20  million  acres  are 
forest  land.     Find  the  area  of  Alabama  in  million  acres. 

37.  In  a  recent  year  the  railroads  of  the  United  States 
owned  70,000  cattle  cars.  Some  of  these  were  single-decked, 
and  others  double-decked.  There  were  44,000  more  of  the 
former  than  of  the  latter.  Find  how  many  cars  there  were  of 
each  kind. 

38.  The  average  number  of  sheep  carried  per  deck  is  45 
larger  than  the  average  number  of  calves.  If  a  double-decked 
car  has  the  average  number  of  calves  on  the  lower  deck  and  of 
sheep  on  the  upper  deck,  it  contains  195  animals.  Find  the 
number  of  sheep  and  of  calves. 


332  A   HIGH   SCHOOL   ALGEBRA 

39.  The  average  number  of  inhabitants  per  square  mile  for 
Indiana  is  -J  of  that  for  Iowa,  and  that  for  Ohio  is  32  greater 
than  that  for  Indiana,  and  62  greater  than  that  for  Iowa. 
Find  the  number  for  each  state. 

40.  Lead  weighs  f  f  times  as  much  as  an  equal  volume  of 
aluminium.  A  certain  statuette  of  aluminium  stands  on  a  base 
of  lead.  The  volume  of  the  base  is  twice  that  of  the  statuette, 
and  the  whole  Aveighs  282  oz.  Find  the  weight  of  the  statuette 
and  of  the  base. 

41.  A  man  inherits  $10,000.  He  invests  some  of  it  in 
bonds  bearing  31  %  interest,  the  rest  in  mortgages  bear- 
ing 5^  %  interest  per  annum.  His  entire  annual  income 
from  these  investments  is  $  510.  Find  the  amount  of  each 
investment. 

42.  A  pile  of  boards  consists  of  inch  boards  and  half-inch 
boards.  There  are  80  boards  and  the  pile  is  58  in.  high. 
How  many  boards  of  each  thickness  are  there  ? 

43.  A  hardware  dealer  sold  a  furnace  for  $  180  at  a  gain  of 
20  %.     What  did  the  furnace  cost  him  ? 

44.  A  merchant  sold  a  damaged  carpet  for  $  42.50  at  a  loss 
of  15  %.     What  did  the  carpet  cost  him  ? 

45.  A  collector  remitted  $  475  after  deducting  from  the 
amount  collected  a  fee  of  5%.  How  many  dollars  did  he 
collect  ? 

46.  The  amount  of  a  certain  principal  at  4  %  simple  interest 
for  1  yr.  was  $  416.     What  was  the  principal  ? 

47.  Pythagoras  being  asked  the  time  of  day,  replied: 
"  There  remains  of  the  day  (from  6  a.m.  to  6  p.m.)  twice  the 
number  of  hours  already  passed.'^     What  time  was  it  ? 

48.  The  three  Graces,  carrying  4  apples  each,  met  the  9 
Muses ;  they  gave  each  Muse  the  same  number  of  apples ; 
then  the  Muses  and  Graces  had  equal  shares.  How  many  had 
each  ? 

49.  A  fruit  vender  gave  a  boy  4  dozen  oranges  to  sell  and 
agreed  to  pay  him  |  ^  for  each  orange  sold,  but  demanded  a 


EQUATIONS 


333 


payment  of  2  j^  for  each  orange  eaten ;   the  boy  disposed  of  all 
the  oranges  and  received  25  j^«    How  many  oranges  did  he  eat  ? 

50.  A  robber  in  escaping  from  a  castle  met  a  guard  whom 
he  bribed  with  ^  of  his  plunder;  at  the  next  gate  he  bribed 
another  guard  with  ^  of  the  plunder  remaining ;  at  the  third 
gate  he  bribed  another  guard  with  i  of  the  plunder  remaining ; 
the  robber  then  escaped  with  2000  ducats.  How  many  ducats 
did  he  steal  ? 

51.  A  servant  agreed  to  work  for  £  10  a  year  and  his  livery. 
At  the  end  of  7  months  his  lord  discharged  him,  giving  him 
the  livery  only.     How  many  pounds  was  the  livery  worth  ? 

SPECIAL  PROBLEMS 

476.  The  solution  of  many  problems  is  made  easier  by  a 
special  plan  of  work. 

EXAMPLE 

How  much  water  must  be  added  to  a  20  %  solution  of  am- 
monia to  make  a  10  %  solution  ? 

Plax.  1.  Consider  an  arbitrary  quantity  of  the  given  mixture  ;  for 
example,  1  gallon. 

2.    Let  X  be  the  number  of  gallons  added  ;  then  the  two  quantities  are : 


1st  Quantity 

2d  Quantity 

Igal. 

(l-fx)  gal. 

3.    Decide  which  substance  (the  ammonia  in  this  problem)  has  not 
changed  in  quantity  ;  state  the  amount  in  each  solution.     Thus, 


Ammonia  in  Ist  Solution 

Ammonia  in  2d  Solution 

20%  Of  Igal. 

10%of  (l+x)gal. 

4.  .-.  the  equation  is  20 7o  of  1  =  10%  of  (1  +  x),  or  .2  =  .1  (1  +  x). 

5.  Therefore,  x  =  1,  and  1  gallon  of  water  must  be  added  for  each 
gallon  of  the  original  solution. 


334  A   HIGH   SCHOOL   ALGEBRA 

WRITTEN    EXERCISES 

Solve  the  following,  using  the  tabular  plan  given  above : 

1.  How  .much  water  must  be  added  to  a  95  %  solution  of 
alcohol  to  make  an  80  %  solution  ? 

2.  A  certain  paint  consists  of  equal  parts  of  oil  and  pig- 
ment. How  much  oil  must  be  added  to  a  gallon  of  this  paint 
to  make  a  paint  f  of  which  is  oil  ? 

3.  How  much  potash  must  be  added  to  a  10  %  solution  to 
make  a  20  %  solution  ? 

4.  Spirits  of  camphor  is  camphor  gum  dissolved  in  alcohol. 
How  many  ounces  of  camphor  gum  must  be  added  per  ounce 
to  a  5  %  solution  in  order  to  make  an  8  %  solution  ? 

477.  Problems  involving  the  rates  of  two  moving  bodies 
have  received  much  attention  in  mathematics. 

For  example,  a  courier,  or  messenger,  leaves  the  rear  of  an  army,  5 
miles  long,  to  deliver  a  dispatch  to  the  officer  at  the  front,  and  rides  at  the 
rate  of  10  miles  per  hour.  10  minutes  later  he  is  followed  by  another 
messenger  riding  at  the  rate  of  15  miles  per  hour.  If  the  army  is  not 
moving,  where  will  the  second  messenger  overtake  the  first  ? 

The  following  are  further  examples  commonly  known  as 
"  clock  "  problems  and  '^  planet "  problems. 

EXAMPLE 

At  what  time  between  3  and  4  o'clock  are  the  hands  of  a 
clock  together  ? 

Solution.     1.    The  minute  hand  moves  1  minute  space  per  minute. 

2.  The  hour  hand  moves  ^^  of  a  minute  space  per  minute. 

3.  Let  X  =  the  number  of  minutes  after  3  o'' clock  when  the  hands  are 
together.  Then,  x  is  the  number  of  spaces  moved  by  the  minute  hand 
and  —  is  the  number  of  spaces  moved  by  the  hour  hand. 

4.  When  they  are  together  the  hour  hand  is  15  +  -^  minute  spaces 
from  XII,  and  the  minute  hand  is  x  spaces. 

5.  Hence,  a;  =  15  +  — ,  and  x  =  163^,  the  number  of  minutes  past 
3  o'clock.  ^^ 


EQUATIONS  335 

WRITTEN    EXERCISES 

1.  How  many  minute  spaces  must  the  minute  hand  gain  on 
the  hour  hand  from  the  time  they  meet  until  they  lie  opposite 
to  each  other  in  the  same  straight  line  ?  At  what  time  are  the 
hands  of  a  clock  opposite  to  each  other  for  the  first  time  after 
12  o'clock  ? 

2.  At  what  time  between  7  and  8  o'clock  are  the  hands 
of  a  clock  opposite  each  other  ? 

3.  At  how  many  different  times,  and  when,  are  the  hands 
of  a  clock  at  right  angles  between  4  and  5  o'clock  ? 

4.  A  and  B  enter  a  race  together ;  at  the  end  of  5  minutes 
A  is  900  yd.  from  the  starting  line  and  75  yd.  ahead  of  B ; 
at  this  point  he  falls,  and  though  he  renews  the  race,  his  rate 
is  20  yd.  a  minute  less  for  the  rest  of  the  course ;  he  crosses 
the  line  ^  minute  after  B.     How  long  did  the  race  last  ? 

5.  In  astronomy  it  is  important  to  know  when  planets  are 
in  line  between  the  earth  and  the  sun.  This  is  called  con- 
junction. Taking  the  earth's  time  of  revolution  about  the  sun 
as  365  days  and  that  of  Venus  as  225  days,  how  long  after  one 
conjunction  of  Venus  until  the  next  one  occurs? 

Suggestion.     The  problem  is  quite  analogous  to    • 

that  of  the  hands  of  a  watch.  For  the  purposes  of 
this  problem  we  suppose  all  the  planets  to  move  in  the  same  plane  and 
in  circular  paths  (orbits)  in  the  same  direction  of  revolution  about  the 
sun  as  a  center. 

1.  Let  X  =  the  number  of  days. 

2.  Venus  will  have  made  -^  revolutions. 

225 

3.  The  earth  will  have  made  —  revolutions. 

365 

4.  But  to  be  in  conjunction,  Venus  (which  goes  faster)  must  have 
made  one  more  revolution  than  the  earth. 

Hence,  JL  =  JL  +  i 
'  225     365        ' 

and  X  =  586||. 

6.    Taking  88  days  as  Mercury's  time  of  revolution  about 
the  sun,  how  long  from  one  conjunction  of  Mercury  to  the  next  ? 


> 


336  A    HIGH   SCHOOL   ALGEBRA 

7.    When  the  earth  is  between  a  planet  and  the  sun,  in  the 
same  line  with  them,  the  planet  is  said  to  be  in  opposition  to 

the  sun.  Taking  687  days  as  Mars' 
time  of  revolution  about  the  sun, 
how  long  is  it  from  one  opposition 
of  Mars  to  the  next  ? 

8.  Taking  4307  days  as  Jupiter's 
time  of  revolution  about  the  sun,  how  long  is  it  from  one 
opposition  of  Jupiter  to  the  next  ? 

9.   Answer  the  same  question  for  Saturn,  whose  time   of 
revolution  is  28.5  yr. 

10.   Also  for  Neptune,  whose  time  of  revolution  is  165.5  yr. 

Note.  Those  who  have  studied  geometry  may  take  up  here  some  of 
the  problems  based  upon  geometric  properties  found  in  Chapter  XXXIH. 

EQUATIONS  WITH  TWO  UNKNOWNS 

478.  Systems  of  Equations.  Two  or  more  equations  con- 
sidered together  are  called  a  system  of  equations. 

479.  Simultaneous  Equations.  Two  or  more  equations  are 
said  to  be  simultaneous  when  all  of  them  are  satisfied  by  the 
same  values  of  the  unknowns. 

480.  All  systems  of  two  independent  simultaneous  equations 
of  the  first  degree  in  two  unknowns  can  be  solved  by  the 
method  of  addition  and  subtraction,  which  consists  in  multiply- 
ing one  or  both  of  the  given  equations  by  such  numbers  that 
the  coefficients  of  one  of  the  unknowns  become  numerically 
equal.  Then  by  addition  or  subtraction  this  unknown  is  elimi- 
nated, and  the  solution  is  reduced  to  that  of  a  single  equation. 

481.  Occasionally  the  method  of  substitution  is  useful.  This 
consists  in  expressing  one  unknown  in  terms  of  the  other  by 
means  of  one  equation  and  substituting  this  value  in  the  other 
equation,  thus  eliminating  one  of  the  unknowns. 

This  may  be  the  shorter  method  when  an  unknown  in  either 
equation  has  the  coefficient  0,  -f  1,  or  —  1. 


EQUATIONS 


337 


General  Solution.     A  general  form  for  two  equations  of  the 
first  degree  is 

ax  +  hy^  e,  (1) 

cx  +  dy=f.  (2) 

From  these  it  is  possible  (without  knowing  the  values  of  a, 
h,  c,  d,  e,  f)  to  find  a  general  form  for  the  solution,  namely : 


^de-hf 
ad  — be' 


y 


af—  ce 
ad  —  be 


(3) 


These  are  the  formulas  for  the  roots  of  any  two  independent 
linear  simultaneous  equations  with  two  unknowns. 

482.   Relation  of  the  Roots  to  the  Constants  in  the  Equations. 

1.    The  denominator  is  the  same  in  each  result  and  is  made 
up  from  the  coefficients  as  follows : 


'ax  -\-by  =  e. 


[cx-{-dy=f. 


Coefficients  ef  x 
a 


2.    The  numerator  of  the  value  of  x  is  made  up  thus 
ax-\-by  =  q. 


Absolute  Terms 


cx-\-dy=f 

3.    The  numerator  of  the  value  of  y  is  made  up  thus : 
ax  -\-by  =  e. 


Coefficients  of  x 
a 


Absolute  Terms 
e 


cx  +  dy  =/. 


338 


A  HIGH   SCHOOL   ALGEBRA 


The  following  examples  will  illustrate  the  use  of  these  for 
mulas  in  solving  equations  : 
2  a;  -  3  2/  =  4, 


1.    Solve 


[^x-\-2y  =  l. 


4x/-3 

3 

2 


y  = 


2\^4 
4/\l 


4.2 


1(_3)   ^  8  +  3  ^11 


2.  2 -(-3).  4     4  +  12 


Test. 


2^-3 

4/\     2 

2.11      3. 


16 


2.1-4.4 

2.2-4(-3) 


^  =  4 

16 


16  ^  -  14  ^  -  7 
16  8 


16 


4.  11      2.-7 
16  8 


16 
16 


W 


(3) 


(4) 


483.  The  above  form  of  expressing  cross  products  is  derived 
from  the  Determinant  Notation,  and  while  it  is  not  necessary 
to  know  Determinants  in  order  to  solve  such  simultaneous 
equations  by  inspection,  it  is  well  to  know  the  basis  of  the 
method. 

a     c 


The  symbol 


b     d 
a  Determinant  of  the  second  order. 


is  defined  to  mean  ad  —  he,  and  is  called 


EXAMPLES 
2-1 


[g    ^1  =  2.5-3. 4=-2,  and   i      ^ 


=  2.5-(-3.  -1)=13. 


ORAL     EXERCISES 


Find  the  value  of* 
3         2 
5         4' 


1. 


2. 


3. 


2 
1     - 

4 

-2 


5. 


6. 


7. 


1 

2" 

2 

4 

2 

a 

3 

h 

a 

b 

X 

y 

EQUATIONS 


339 


484.   The  values  of  x  and  y  in  Section  .481  may  be  expressed 
in  Determinant  Notation. 


Thus, 


Solve 


j4a;  +  72/ 
•      I    x-2y  =  12 


e    h 

a    e 

f    d 

y  = 

c    f 

a    b 

a    b 

c    d 

c    d 

EXAMPLE 

l  =  — 

27, 

-27         71 

12     -.2L -27.  (-2)- 7 -12  ^-30 
14         71  4.  (-2) -7  -15 

1     -2 


2. 


4 

-27 

1 

12 

4 

7 

1 

-2 

4.12-l(-27)_    75 
-  15  ~  -  15 


Test.     4  •  2  +  7(- 5)  =  - 27  ;   2  -  2(- 5)=  12. 


-5. 


(2) 
(3) 


(4) 


485.  This  method  usually-  gives  the  values  by  inspection, 
for  the  products  and  differences  can  be  read  direct  from  the 
equations  themselves. 


For  example : 
4:x  +  2y  =  l, 
3x  —  5y  =  4:. 

Denominators 

=  4(-5)-2 
=  -26. 


4:X  +  2y  =  l, 
3  a;  —  5  2/  =  4. 
Numerator  of  y 
=  4  .  4  -  1  .  3  =  13. 
13    ^     1 
-26  2* 


y  = 


4.x  +  2y  =  l, 
Sx  —  5y  =  4:. 

Numerator  of  x 
=  1(_5)_2.4=-13. 

-26      2 


WRITTEN     EXERCISES 

Solve,  using  determinant  forms  : 

1.  x-{-y  =  5,  3.   2x  +  y  =  8, 
x  —  y  —  8.  x  +  y  =  2. 

2.  aj  +  ?/  =  5,  4.    2  a;  4- 3?/ =  7, 
x  —  y  —  1,  x  —  y  =  l. 


5.  2x-h2^  =  8, 
2x-y=:2. 

6.  3x  —  y  —  —  5, 
2x-y  =  -3, 


340  A  HIGH   SCHOOL  ALGEBRA 

7.  4:X-Sy  =  7,  23.   4tx-7y  =  5j 

3  a;  -  4  2/  =  7.  Sx-\-15y  =  39. 

8.  5x-^y  =  9,  24.   2x-y=:6, 
Sx-\-y  =  5.  10x-3y=.7. 

9.  4:X-\-5y  =  22,  25.   7x-\-9y  =  20, 
Sx  +  2y  =  lS.  Sx-4.y  =  ~20. 

10.  ic-52/  =  -22,  26.   2^x-6y  =  6, 
5x-y=:10.  4.x-^2y  =  16. 

11.  4.x-Sy  =  S,  27.   2a;  +  2/  =  0, 
3a;-42/=-3.  a;-t-2?/  =  -3. 

12.  4.x  +  2y  =  l,  28.    11  a; -f  22 2/ =  33, 
3aj-2^  =  |.  4.x  +  lSy  =  22. 

13.  12aj-ll2/  =  87,  29.   1.4 a; +  2.1 2/ -1, 
4a;  +  22/  =  46.  2.8  a^  +  3.3 2/ =  2. 

14.  7aj-22/  =  3,  30.   30 a^  +  25 2/ =  40, 
7aj-4'2/  =  -l.  13  a; +  16  2/ =  221. 

16.  9a;-32/  =  -6,  ^^'    9a.  -  12.y  =  - 51, 
8a.-22/  =  -6.  21a.-352/=-133. 

16.  a.+2/=i,  ,  ''-'r'r'i'^ 

6a.  +  2/  =  2.  10a:-52/=15. 

33.  5  a;  =  3  2/, 

17.  aa.  +  62/  =  c,  2a.  +  82/  =  4. 
px-\-qy  =  d. 

34.  ax—oy  =  c, 

18.  a;-TO2/  =  a,  «  +  a2/  =  6. 
a;  +  »?/  =  6. 

^^  35.   4a2aj  +  5a2/  =  3, 

19    ax-hy  =  e,  Qax  +  7y  =  2. 

cx  —  dy=f.  „„          ,  , 

^     -^  36.    ax  +  by  =  c, 

20.  ax-y  =  b,  a'x  +  hhj  =  c\ 
^^  +  y  =  ^'  37.    c.  +  62/  =  l, 

21.  2a:  +  72/  =  ll,  «^4.,,_q 
5aj-92/  =  l.  a^^""^' 

22.  3a;  +  72/  =  -l,  38.   |aj  +  |2/  =  i, 
2a;-32/  =  7.  |«'  +  A2/  =  4. 

Note.    See  Chapter  XXXIII  for  problems  relating  to  geometry. 


EQUATIONS 


341 


SPECIAL  SYSTEMS 

486.  The  study  of  graphs  of  systems  of  equations  helps  to 
interpret  special  cases. 

487.  We  have  solved  (Sec.  481)  the  equations 

ax  -\-hy  =  e, 
cx  +  dy=f, 

and  found  that 


^^de-zM.  y=  «/-ec 


ad  —  be 


ad  —  be 


I.  Let  us  give  to  the  letters  a,  b,  c,  d,  such  values  that 
ad—bG  =  0;  for  example,  a  =  2,  b  =  l,  c  =  4,  d  =  2.  And  let 
us  give  to  e  and  /  such  values  that  de  —  bf  is  not  0 ;  for  example, 
e  =  5,/=4. 

Then  the  above  results  become : 

X=-'       =-^ 

The  indicated  division  by  zero  means  that  the  solution  is  impossible,  Sec. 
430.  There  is  no  pair  of  values  that  satisfies  both  equations.  This  appears 
readily  also  by  substituting  the  values  2,  1,  4,  2,  6,  4,  for  a,  6,  c,  d,  e,/, 
in  the  given  equations,  which  then  become : 

2x  +  y  =  5,\  f2x  +  ?/  =  5, 

4x+2y  =  4:,\    "      \2x+y  =  2. 
It  is  obvious  that  no  set  of  values  of  x  and  y  can  make  2x  +  y  equal  to  5 
and  also  equal  to  2. 

This  condition  can  be  illustrated 
graphically : 

For  drawing  the  graphs  of 

2x  +  y  =  5  and  2x-\-y  =  2, 

the  two  lines  are  parallel.  That  two  paral- 
lel straight  lines  do  not  intersect  is  the  ge- 
ometric condition  corresponding  to  the  fact 
that  a  system  of  two  incompatible  equations 
has  no  solution. 


The  two  equations  are  called  incompatible  or  contradictory. 


342 


A  HIGH   SCHOOL  ALGEBRA 


II.  Retaining  the  values  of  a,  b,  c,  d,  above,  let  us  give  e  and 
/  such  values  that  the  numerators  of  the  result  both  become 
zero;  for  example,  e  =  5,/=  10. 

The  result  assumes  the  form  : 

0  0 

Under  these  conditions  we  have  seen,  Section  430,  that  x  and  y  may- 
have  any  values.     But  this  may  also  be  seen  by  reference  to  the  equations. 
Substituting  the  values  2,  1,  4,  2,  5,  10,  for  a,  6,  c,  d,  e,  /,  in  the  given 
equations,  they  become  : 

2x+y  =  b. 
4  a;  +  2  y  =  10. 


It  appears  that  the  second  equation  is  twice  the  first,  and  hence  equiva- 
lent to  it.  Any  values  of  x  and  y  that  satisfy  the  first,  will  also  satisfy 
the  second. 

We  can  choose  arbitrarily  any  value  for  x  and  then  determine  a  value 
of  y  to  go  with  it  by  means  of  the  first  equation.  For  example,  choosing 
X  =  3,  then  2  •  3  +  2/  =  5,  which  gives  y  =—\.  These  values  of  x  and  y 
satisfy  both  equations.  Similarly,  any  value  can  be  chosen  for  y^  and  a 
value  of  x  found  such  that  the  pair  of  values  satisfies  the  given  system. 


The  two  equations  are  dependent. 
a  solution  of  the  other. 


Every  solution  of  one  is 


If  we  undertake  to  make  the 
graphs  of  the  two  equations  as 
given,  we  find  that  they  lead  to 
the  same  straight  line.  The 
two  graphs  are  coincident ;  every 
point  of  the  straight  line  is  a 
common  point  of  the  two  graphs. 
Any  abscissa  x  is  the  abscissa  of 
a  common  point  of  the  graphs ; 
any  ordinate  y  is  the  ordinate  of 
a  common  point  of  the  graphs. 

Note.      The  study  of  expres- 
j      i       :      :       i      i      i       i       i  sions  which  may  assume  the  ex- 

^      '-^      '••'■•'      i  ceptional  forms  mentioned  above, 

especially  those  which  may  assume  the  form  - ,  is  very  important,  both 

from  the  point  of  view  of  later  mathematics  and  the  physical  sciences. 


EQUATIONS  343 

488.  Number  of  Solutions.  We  have  thus  seen  that  systems 
of  two  linear  equations  in  two  unknowns  may  be  classified  as 
follows : 

1.  Independent  (the  ordinary  case,  admitting  one  solution). 

2.  Contradictory  (admitting  no  solution). 

3.  Dependent  (admitting  a  boundless  number  of  solutions). 

WRITTEN     EXERCISES 

Construct  the  graphs  of  each  of  the  following  systems  and 
classify  them  according  ta  Section  488 : 

1.  ^x-{-y  =  2,  4.   a;  +  2  2;  =  10,  7.   7  a;  +  14  2/  =  7, 
x+y=^0.               x-{-Zz  =  ll,  x-{-2y=2. 

2.  2x-y=l,  5.   x  =  25,  8.   12x-8y=S, 
4:X-2y  =  2.         y  =  10.  Sy-x  =  4:. 

3.  s  —  t  =  6,  e.   10x+  5y  =  25y  d.   2x  —  Sy  =  —  5, 
s-\-t=6,                2x+y  =  5  x-\-2y  =  S. 

EQUATIONS   WITH   THREE   OR   MORE   UNKNOWNS 

489.  The  definitions  and  methods  for  the  solution  of  two 
equations  with  two  unknowns  may  be  applied  equally  well  to 
a  greater  number  of  equations  and  unknowns. 

To  solve  three  linear  equations  with  three  unknowns,  elimi- 
nate one  unknown  from  any  pair  of  the  equations  and  the 
same  unknown  from  any  other  pair;  two  equations  are  thus 
formed  which  involve  only  two  unknowns  and  which  may  be 
solved  by  methods  previously  given. 

Four  or  more  equations  with  four  or  more  unknowns  may 
be  solved  similarly. 

490.  Determinants  of  the  third  and  higher  orders  have  been 
defined,  and  can  be  used  to  solve  linear  equations  with  three  or 
more  unknowns,  but  the  method  is  too  complicated  to  be  of 
practical  value  here. 


4. 

8aj-2 

2aj  +  3 
42/  +  7 

z-^5  = 
y-21 

2-69: 

:0, 

=  0, 
=  0. 

5. 

y-i-z  = 
x-\-z  = 

1 
1 

1. 

c 

344         A  HIGH  SCHOOL  ALGEBRA 

WRITTEN    EXERCISES 

Solve : 

1.  4.x-2y-\-z  =  S, 
x-\-Sy-\-2z  =  13,  ^ 
-Sx-\-12y-\-z=:21. 

2.  5x  +  4:y-{-2z  =  17, 
Sx-2y-\-5z  =  2, 
2x-y-{-3z  =  2. 

3.  x  —  y  —  z  =  a, 
Sy  —  x  —  z  =  2a, 
7z  —  y  —  x  =  4:a. 

6.  -^  +  -^  +  ^=2c, 
a  +  0     o  —  ca-^c 

-^ y ^  =  2a, 

a—b     b—c     a—c 

^  +  J '—  =  2a-2c. 

a  —  b     G  —  b     a-\-c 

7.  X  —  y  —  z  —  2  10  —  —  12, 
3a;-2/-22  +  8w  =  40, 
4aj  —  42/  +  72;  —  5w  =  52, 

Sx  —  y-\-2z-\-w  =  4:4:. 

8.  A  and  B  can  do  a  piece  of  work  in  6  days ;  A  and  C  can 
do  it  in  9  days,  and  A,  B,  and  C  can  do  8  times  the  whole  work 
in  45  days.     In  how  many  days  can  each  do  it  alone  ? 

9.  A  sum  of  money  is  divided  into  3  parts  such  that  the 
first  part  exceeds  the  second  part  by  $  100.  The  annual  in- 
come from  the  second  and  third  parts,  if  invested  at  6  %  per 
annum,  is  $42.  The  sum  of  the  first  and  second  parts  equals 
the  sum  of  the  second  and  third  parts.  Find  the  number  of 
dollars  in  each  part. 

Nio.  A  number  consists  of  3  digits.  If  99  is  added  to  it  the 
sum  is  a  number  having  the  same  digits,  but  in  reverse  order. 
The  sum  of  the  hundreds'  and  tens'  digits  equals  the  units' 
digit,  and  the  units'  digit  exceeds  the  tens'  digit  by  1. 


EQUATIONS  345 

11.  The  sum  of  the  digits  in  a  certain  number  of  three  figures 
is  13,  the  hundreds'  digit  exceeds  the  tens'  digit  by  1,  and  the 
units'  digit  exceeds  the  hundreds'  digit  by  2.    Find  the  number. 

12.  Three  casks  together  contain  79  gallons  ;  the  second  con- 
tains 3  gallons  more  than  i  as  much  as  the  first,  and  the  third 
contains  7  gallons  less  than  the  second.  How  many  gallons 
are  there  in  each  ?     (From  a  fourteenth  century  manuscript.) 

Note.  Those  who  have  studied  geometry  may  take  up  here  some  of  the 
problems  based  upon  geometric  properties  found  in  Chapter  XXXIII. 

QUADRATIC  EQUATIONS 

491.  Quadratic  Equations.  Equations  of  the  second  degree 
are  called  quadratic  equations. 

A  general  form  for  quadratic  equations  in  one  unknown  is 

acc^  +  bx  +  c  =  0, 

in  which  a,  b,  c  represent  any  known  numbers,  except  that  a 
may  not  be  zero. 

492.  Solution  of  Quadratic  Equations. 

(1)  The  incomplete  quadratic  equation  x^  =  a  is  solved  by 
extracting  the  square  root  of  both  members.  The  roots  are : 
x=  ±  Va. 

(2)  The  incomplete  quadratic  equation  aoi:^-\-bx  =  Ois  solved 

by  factoring.     The  roots  are  a?  =  0  and  x  = 

a 

(3)  Complete  quadratic  equations  are  solved  by  completing 
the  square. 

The  process  consists  of  two  main  parts  : 

(a)  Making  the  left  member  a  square  while  the  right  member 
does  not  contain  the  unknown. 

This  is  called  completing  the  square. 

It  is  based  upon  the  relation  (x  +  ay  =  x^  +  2  ax -h  a^,  in  which  it  ap- 
pears that  the  last  term,  a^^  is  the  square  of  one  half  of  the  coefficient  of  x. 

(b)  Extracting  the  square  roots  of  both  members  and  solving 
the  resultiyig  linear  equations. 


346  A   HIGH   SCHOOL  ALGEBRA 

Square  roots  whicli  cannot  be  found  exactly  should  be  indi- 
cated. 

EXAMPLE 


ve: 

x'-Sx-^-d^O. 

y) 

Transposing, 

x2  _  8  X  =  -  9. 

(2) 

Completing  the  square? 

a;2-8a;  +  16=-9  +  16. 

(5) 

Kearranging, 

(a;  _  4)2  =  7. 

(-#) 

Extracting  the  square  root, 

x-4=±V7. 

(5) 

Solving  (5)  for  x. 

x  =  4±V7. 

(6) 

(4)  If  any  quadratic  equation  has  zero  for  the  right  member, 
and  if  the  polynomial  constituting  the  left  member  can  be  fac- 
tored, the  quadratic  is  equivalent  to  two  linear  equations  whose 
roots  can  readily  be  found.     (See  Chapter  XIII.) 

WRITTEN     EXERCISES 

Solve : 

1.  3a^  =  18.  3.   a^-5a;-f-6  =  0.       5.f-2t-6  =  0. 

2.  sc^-5x  =  0.       4.    a^  +  4a;-3  =  0.        6.    Sp^  =  5p. 

7.  a^  +  lla;-t-24  =  0.  ^^     ^=  _i_ +  ?^_^  1  35 

8.  5a;2-13a;  +  5  =  0.  *    4     x-S       5 

9.  15  2/' +  134  2/ +  288  =  0.       11.    -J— +  -1 i-  =  0. 


y-S     1-2/     y-2 


-<,     6  —  2ic.5-\-w     w  —  5 


w  —  2       3  +  w;     2  —  w 

13.  7  (7  -  z)(z  -  6)  +3  (5  -  z)(2  -z)-^0  =  0. 

14.  Find  two  numbers  whose  sum  is  10,  and  the  sum  of 
whose  squares  is  68. 

Suggestion.     Let  x  represent  one  number,  and  10  —  a;  the  other. 

15.  A  room  is  3  yd.  longer  than  it  is  wide;  at  $1.75  per 
square  yard,  carpet  for  the  room  costs  $  49.  Find  the  dimen- 
sions of  the  room. 

16.  A  man  bought  for  $300  a  certain  number  of  oriental 
rugs,  each  at  the  same  price.  If  he  had  bought  rugs  each 
costing  $  40  more,  he  would  have  obtained  2  fewer  rugs.  How 
many  rugs  did  he  buy  ? 


EQUATIONS  347 

17.  A  dealer  bought  a  number  of  similar  tables  for  $153. 
He  sold  all  but  7  of  them  at  an  advance  of  $  1  each  on  their 
cost,  thus  receiving  $  100.     How  many  tables  did  he  buy  ? 

18.  A  man  invested  $6000  at  a  certain  rate  of  simple 
interest  during  4  years.  At  the  end  of  that  time  he  reinvested 
the  capital  and  the  interest  received  during  the  4  years  at  a 
rate  of  interest  1  %  lower  than  at  first.  His  annual  income 
from  the  second  investment  was  $  372.  What  was  the  original 
rate  of  interest  ? 

19.  A  rectangle  whose  area  is  84  sq.  in.  is  5  in.  longer  than 
it  is  wide.     Find  its  dimensions. 

20.  A  certain  number  of  men  hire  an  automobile  for  $  156. 
Before  they  start,  two  others  join  them,  sharing  equally  in  the 
expense.  The  amount  to  be  paid  by  each  of  the  original  renters 
is  thus  reduced  by  $13.     How  many  men  were  there  at  first? 

21.  A  man  rows  down  a  stream  a  distance  of  21  mi.  and 
then  rows  back.  The  stream  flows  at  3  mi.  per  hour  and  the 
man  makes  the  round  trip  in  13|-  hours.  What  is  his  rate  of 
rowing  in  still  water  ? 

22.  The  product  of  a  number  and  the  same  number  increased 
by  40  is  11,700;  what  is  the  number? 

23.  If  each  side  of  a  certain  square  is  increased  by  5  the 
area  becomes  64 ;  what  is  the  length  of  a  side  ? 

24.  Find  two  numbers  whose  sum  is  16  and  the  difference 
of  whose  squares  is  32. 

25.  A  number  multiplied  by  5  less  than  itself  is  750.  Find 
the  number. 

26.  The  product  of  two  consecutive  even  numbers  is  728. 
Find  the  numbers. 

INTERPRETATION   OF  RESULTS 

493.  Interpretation  of  Results.  After  the  conditions  of  a 
problem  have  been  expressed  by  equations,  and  the  equations 
solved,  the  result  must  be  examined  to  see  whether  it  is 
admissible  under  the  conditions  of  the  problem. 


348  A   HIGH   SCHOOL   ALGEBRA 

EXAMPLES 

1.  Find  three  consecutive  integers  such  that  their  sum  shall 
be  equal  to  3  times  the  second. 

Solution.     1.  Let  %  =  the  first. 

2.  Then,  ic  +  1  =  the  second, 

3.  and  x  +  2  =  the  third. 

4.  ...  x+(a;  +  l)  + (a;  +  2)  =  3(x  +  l). 

5.  .-.  (3  -  3)(x  +  1)  =  0,  or  0(x  +  1)  =  0. 

Interpretation  of  the  Result.  The  equation  determines  no  par- 
ticular value  of  a: ;  it  exists  for  every  value  of  x.  Consequently,  every 
three  consecutive  integers  must  satisfy  the  given  conditions. 

2.  Find  three  consecutive  integers  whose  sum  is  57,  and  the 
sum  of  the  first  and  third  is  40. 

Solution.     1.  Let  x  =  the  first. 

2.  Then  x  -\-  1  =  the  second, 

3.  and  x  +  2  =  the  third. 

4.  Then,  x -\-  (x -\-l)  +  (x +  2)  =  57, 

5.  and  x  ■}-  (x  +  2)  =  40,  by  the  given  conditions. 

6.  From  (4),  x  =  18. 

Interpretation  of  the  Result,  cc  =  18  v^ill  not  satisfy  equation  (5); 
therefore  no  three  consecutive  integers  satisfy  the  problem. 

3.  The  town  B  is  d  mi.  from  A ;  two  trains  leave  A  and  B 
simultaneously,  going  in  the  same  direction  (that  from  A 
towards  B),  A  at  the  rate  of  m  mi.  per  hour  and  B  q  mi.  per 
hour.     How  far  from  B  will  the  trains  be  together  ? 

Solving  this  problem  by  the  usual  method,  we  find  as  the  result     ^      . 

m  —  q 

Interpretation  of  the  Result.     If  d  is  not  equal  to  0,  and  if  m  =  g, 

the  result  assumes  the  form  ^  .    This  means  that  the  problem  is  impossible 

under  these  conditions.  This  is  evident  also  from  the  meaning  of  m  and 
q  in  the  problem.  If  the  two  trains  go  in  the  same  direction  at  the  same 
rate,  the  one  will  always  remain  d  miles  behind  the  other. 

If,  however,  d  =  0,  and  m  =  g,  the  result  assumes  the  form  -,  which 

equals  any  number  whatever.  This  also  agrees  with  the  conditions  of  the 
problem.  If  d  is  zero,  B  and  A  are  coincident,  and  the  two  trains  are 
together  at  starting.  If  m  =  g,  they  both  run  at  the  same  rate,  and  always 
remain  together.     They  are  therefore  together  at  every  distance  from  B. 


EQUATIONS  349 

WRITTEN   EXERCISES 
Solve  and  interpret  the  results  : 

1.  Fifteen  clerks  receive  together  $150  per  week;  suppose 
that  some  receive  $8  and  others  $12  per  week.  How  many- 
would  there  be  receiving  each  salary  ? 

2.  A  train  starts  from  New  York  for  Richmond  via  Phila- 
delphia and  Baltimore  at  the  rate  of  30  miles  an  hour,  and 
two  hours  later  another  train  starts  from  Philadelphia  for 
Richmond  at  the  rate  of  20  miles  an  hour.  How  far  beyond 
Baltimore  will  the  first  train  overtake  the  second,  given  that 
the  distance  from  New  York  to  Philadelphia  is  90  miles  and 
from  Philadelphia  to  Baltimore  96  miles  ? 

3.  If  the  freight  on  a  certain  class  of  goods  is  2  cents  per 
ton  per  mile,  together  with  a  fixed  charge  of  5  cents  per  ton 
for  loading,  how  far  can  2000  tons  be  sent  for  $80? 

4.  Find  three  consecutive  integers  whose  sum  equals  the 
product  of  the  first  and  the  last. 

5.  The  hot-water  faucet  of  a  bath  tub  will  fill  it  in  14 
minutes,  the  cold-water  faucet  in  10  minutes,  and  the  waste 
pipe  will  empty  it  in  4  minutes.  How  long  will  it  take  to  fill 
the  tub  when  both  faucets  and  the  waste  pipe  are  opened? 

REVIEW 

WRITTEN  EXERCISES 

Solve : 

1     x  +  ^^x-1  ^     a;  + 150  ^  6 

x-l~x-^s'  '     x-^50      5* 

2.  aj2-14x-f33  =  0.  8.    4x-M9  =  5a;-l. 

3.  a  +  -=c.  c    A       ^       r 

X  9.   4--— =6. 

4.  2x+S(4.x-l)  =  5(2x-{-7). 

.     i.         ^/o     /a^      q/    .in     10.   3  a; -40  =  100--. 

5.  14ic  — 5(2a;4-4)  =3(a;+l).  2 

3x_ix^^  ^_2_^  +  ?  =  24. 

2        3  2  14^7 


85( 

)                       A   HIC 

12. 

T-^  =  T-^- 

13. 

2x     ^•'-^-35. 

A   HIGH   SCHOOL  ALGEBRA 

22.  120_i20^^2 

X        x-{-3 

23.  x(l  -x)=^12. 
^                                  24.  aj2_8a;  =  9, 

14.  1^4-875  =  4000.  ^  r  7         ^         k 

16  25.  ^Jli-  +  -^_=£ 

15.  1  x  +  A:-lox  =  Qx-^2.  ^         ^^^      ^ 

16.  5^  +  4  =  9^  +  1.  2^-  2/^  +  32/  =  70. 

27.  z^-Sz  =  70. 


X  7 


a;  +  60     3x-5 

18. 

2(2a-bx)_a 
3  62            5 

19. 

180        a? 

2£c  +  6     2 

20. 

100a;-a^  =  -2400. 

21. 

a;(a;4-4)=45. 

28.    11  m2  -  10  m  =  469. 
23     90_    27  90 


r      r +2      r +1 
30.    ^+     8  32 


5  —  x     4:  —  x     X  -\-2 

31.    2?/-32/  +  120 

=  42/-62/H-132. 

32.  7a;2-10a;  =  120+4a;2_19a;. 

33.  (5  05  -  2)(6  ic  + 1) -(10  a;  +  3)(3  a;  + 10)  =  0. 

34.  (2a;-3)(a;  +  l)-(3a;-7)(a;-4)=36. 

35.  _^  +  -^=_^_.       38.    ^=7, 
a;4-2      a;-2      2a;-3  y 

onOiK         a  2x  —  10y  =  3y  +  2. 

36.  2  cc  +  5  2/  =  6,  ^  "^ 

4x  +  lly  =  S,  33     7y-6^1 

37.  5.-3,  =  l,  *    "i^-^t     f 

^  2i»  +  l      3 

.-     3a;  +  2y  +  5  ,  g;-3y-13  ,  y-Sx-{-S_^ 
40.  -  +  ^  +  g  -6, 

2a;  — 4  ?/4-6_  _  « 
?/  — 5  a;-f- 11 
41.    6^  =  43  —  52/,  42.    3a;  +  4?/  +  2  2  =  — 4, 

32  =  37  —  4  a!,  2x—  5y—  z  =  9, 

42/  =  55 -5  a;.  -4a;4-22/  +  3;2  =  -23. 


43. 


EQUATIONS  351 

y  —  2_ 1     X— 2_2     x  —  y_l 

x-\-y     5'    y  +  z     S^    x-^z      4* 
44.    (z-2)(x-\-S)  =  (x-r)(z-l), 
(z-{-S)(y-2)-(y  +  2)(z  +  2)  =  0, 
yi^3-2x)-{-(2y-S)(l  +  x)=0. 

45  3  3a;^  +  14      1  +  3a;     ^^^q 

'    2 a?  + 10      7(4:-x)        5-^x        7 

46.    2(2a;  +  32/)-^M)-^=9,  ^  +  2/  =  l. 
o  4 

47.  A  rectangle  whose  length  is  greater  than  its  breadth  by 
1  yd.  has  an  area  of  6  sq.  yd.     Find  its  dimensions. 

48.  A  square  {2x  —  3)  ft.  on  a  side  has  taken  from  it  a 
square  x  ft.  on  a  side.  The  remaining  area  is  24  sq.  ft. ;  find 
the  side  of  each  square. 

49.  The  product  of  two  consecutive  numbers  is  380;  find 
the  numbers. 

50.  The  product  of  two  consecutive  even  numbers  is  840. 
Eepresent  the  smaller  by  27i  and  find  both  numbers. 

51.  In  a  certain  election  36,785  votes  were  cast  for  the 
three  candidates  A,  B,  C.  B  received  812  votes  more  than 
twice  as  many  as  A;  and  C  had  a  majority  of  one  vote  over 
A  and  B  together.     How  many  votes  did  each  receive  ? 

52.  In  a  certain  election  there  were  two  candidates,  A  and 
B.  A  received  10  votes  more  than  half  of  all  the  votes  cast. 
B  received  4  votes  more  than  one  third  of  the  number  received 
by  A.     How  many  votes  did  each  receive  ? 

53.  A  group  of  friends  went  to  dine  at  a  certain  restaurant. 
The  head  waiter  found  that  if  he  were  to  place  five  persons 
at  each  table  available,  four  would  have  no  seats,  but  by  plac- 
ing six  at  each  table,  only  three  persons  remained  for  the  last 
table.     How  many  guests  were  there,  and  how  many  tables? 

54.  A  flower  bed  of  uniform  width  is  to  be  laid  out  around 
a  rectangular  house  20  ft.  wide  and  36  ft.  long.  What  must 
be  the  width  of  the  bed  in  order  that  its  area  may  be  one 
third  of  that  of  the  ground  on  which  the  house  stands? 


352  A   HIGH   SCHOOL  ALGEBRA 

55.  Wood's  metal,  which  melts  in  boiling  water,  is  made 
up  of  one  half  (by  weight)  of  bismuth,  a  certain  amount  of 
lead,  half  that  much  zinc,  and  half  as  much  cadmium  as  zinc. 
How  much  of  each  is  there  in  100  lb.  of  Wood's  metal  ? 

56.  If  in  the  preceding  exercise  f  as  much  cadmium  as  zinc 
is  used,  a  different  metal  is  formed.  How  many  pounds  of 
each  constituent  metal  in  100  lb.  of  this  metal  ? 

57.  A  cask  contains  10  gal.  of  alcohol.  A  certain  number 
of  quarts  are  drawn  out ;  the  cask  is  then  filled  up  with  water 
and  the  contents  thoroughly  mixed.  Later,  twice  as  many 
quarts  are  drawn  out  as  the  previous  time  and  the  cask  filled 
up  with  water.  There  now  remain  only  4.8  gal.  of  alcohol 
in  the  mixture.     How  many  gallons  were  drawn  out  at  first  ? 

Suggestion.  Let  x  —  the  number  of  gallons  first  withdrawn.  Then 
10  —  a;  =  the  number  of  gallons  left. 

When  the  cask  is  filled  again  with  water  any  part  of  the  mixture  is 

— ^^  alcohol.     Then,  — ^^  of  the  2  x  gallons  of  mixture  withdrawn 
10  '      10  ^ 

the  second  time  is  alcohol.     Hence,  10  —  a;  —  (      ~^]2a;  is  the  number 

of  gallons  of  alcohol  left  in  the  cask.  \    10     / 

58.  How  much  water  must  be  added  to  30  oz.  of  a  6  %  solu- 
tion of  borax  to  make  a  4  %  solution  ? 

59.  How  much  acid  must  be  added  to  10  quarts  of  a  2  % 
solution  to  make  a  5  %  solution  ? 

60.  Alcides  was  asked,  "How  many  are  there  of  your 
numerous  herd  ?  "  He  replied  :  "  If  I  had  6  less  than  twice  as 
many  more,  the  number  would  be  306.     Find  the  number." 

61.  A  courier  went  from  Paris  to  Grenoble,  1'20  leagues,  in 
4  days,  each  day's  journey  being  2  leagues  shorter  than  that  of 
the  preceding  day.  How  many  leagues  did  he  travel  each 
day  ?     (Ozanam's  Algebra,  1702.) 

62.  Two  messengers,  A  and  B,  set  out  towards  each  other 
from  two  places  59  mi.  apart,  B  starting  1  hr.  after  A.  A 
goes  7  mi.  in  2  hr.,  and  B  8  mi.  in  3  hr.  How  far  will  A 
have  gone  when  he  meets  B  ?  (Newton's  Arithmetica 
Universalis,  1707.) 


EQUATIONS  353 

63.  A  merchant  bought  a  certain  number  of  platters  for 
$  366.  Three  were  broken  during  shipment.  He  sold  ^  of  the 
remainder  for  $  75  at  a  profit  of  25  % .  Find  the  number  of 
platters  bought  and  the  price  per  platter. 

64.  A  certain  hall  contains  both  gas  jets  and  electric  lights. 
When  60  gas  jets  and  80  electric  lights  are  used,  the  cost  for 
an  evening  is  $  4.  If  90  gas  jets  and  60  electric  lights  are 
used,  the  cost  is  $4.05.  Find  the  cost  per  gas  jet  and  electric 
light. 

65.  A  tailor  paid  $  12  for  4  yd.  of  cloth  and  8  yd.  of  lining. 
At  another  time  he  paid  $  21  for  6  yd.  of  the  cloth  and  16  yd. 
of  the  lining.     Find  the  price  of  each  per  yard. 

66.  Two  wheelmen  are  328  ft.  apart  and  ride  toward  each 
other.  If  A  starts  3  seconds  before  B,  they  meet  in  14  seconds 
after  A  starts  ;  or  if  B  starts  2  seconds  before  A,  they  meet  in 
14  seconds  after  B  starts.     Find  the  rate  of  each. 

67.  A  man  had  a  portion  of  his  capital  invested  in  stocks 
paying  6  %  dividends,  the  remainder  in  mortgages  paying  5  %. 
His  annual  income  was  $  700.  The  next  year  the  dividend  on 
the  stock  was  reduced  to  5  %,  but  by  reinvestment  he  replaced 
his  old  mortgages  by  new  ones  paying  51%.  His  income  for 
this  year  was  $  690.  How  much  had  he  invested  in  stocks  ? 
Also  in  mortgages  ? 

68.  A  company  at  a  tavern,  when  they  came  to  pay,  found 
that  if  the  same  bill  were  divided  among  three  persons  more, 
the  amount  would  be  one  shilling  less  per  person ;  and,  if  it 
were  divided  among  two  persons  fewer,  it  would  be  one  shilling 
more  per  person.  Find  the  number  of  persons  in  the  original 
company,  and  the  amount  of  the  bill.  (Saunderson's  Algebra, 
1740.) 

69.  One  person  says  to  another,  "  If  you  give  me  three  of 
your  coins,  I  shall  have  as  many  as  you."  The  second  person 
replies,  "  If  you  give  me  three  of  yours,  I  shall  have  twice  as 
many  as  you  have."  Find  the  numbers  that  each  has.  (Oza- 
nam's  Algebra,  1702.) 


354  A   HIGH   SCHOOL  ALGEBRA 

70.  A  coacli  set  out  from  Cambridge  to  London  with  four 
more  passengers  outside  than  within.  Seven  outside  passengers 
could  travel  at  2  shillings  less  expense  than  4  inside  passen- 
gers. The  fares  of  all  the  passengers  amounted  to  180  shil- 
lings. At  the  end  of  half  the  journey  the  coach  took  up  1 
more  inside  and  3  more  outside  passengers ;  these  paid  -f^  as 
much  as  the  others.  Required  the  number  of  passengers  and 
the  fare  of  each.     (Bland's  Algebraical  Problems,  1816.) 

71.  Seven  years  ago  a  man  was  4  times  as  old  as  his  son ; 
7  years  hence  he  will  be  only  double  his  age.  Find  the  age 
of  each.     (Simpson's  Algebra,  1767.) 

SUMMARY 

The  following  questions  summarize  the  definitions  and  pro- 
cesses treated  in  this  chapter : 

1.  State  the  general  form  of  a  linear  equation  with  one  un- 
known, and  the  formula  for  its  root.  Sees.  472-^75. 

2.  When  is  a  system  of  equations  simultaneous  ?     Sec.  479. 

3.  State  the  general  form  of  a  system  of  two  linear  equa- 
tions with  two  unknowns.  Sec.  481. 

4.  State  two  methods  of  solving  simultaneous  equations. 

Sees.  480,  481. 

5.  What  is  the  general  form  of  the  values  of  the  unknowns 
in  simultaneous  equations  with  two  unknowns  ?  Sec.  481. 

6.  What  use  may  be  made  of  determinants  in  solving  simul- 
taneous equations  ?  Sees.  483,  484. 

7.  What  is  the  general  form  of  quadratic  equations  with  one 
unknown  ?  Sec.  491. 

8.  Name  two  kinds  of  quadratic  equations,  and  the  method 
of  solving  each.  Sec.  492. 


CHAPTER  XXVI 

EXPONENTS  AND   ROOTS 

LAWS  OF  EXPONENTS 

494.  Preparatory 

1.  What  is  the  meaning  of  a^  ?     Of  a^  ?     Of  a"  ? 

2.  What  is  the  meaning  of  Va2?     Of  ^a^?     Of  Va"? 

4.  a^-^a^  =  ?     a^-r-a^=?     a^^a^=?     h^^^h^^? 

5.  (a2)2  =  ?     (a2)3  =  ?     (c5)2=?     (a;io)3  =  ? 

495.  We  shall  soon  define  negative  and  fractional  exponents, 
but  until  this  is  done  literal  exponents  are  to  be  understood  to 
represent  positive  integers. 

496.  Law  of  Exponents  in  Multiplication. 

I.  a""  '  0."  =  a'"+^ 

For  a"^  =  a  '  a  •  a  •••  to  m  factors, 

and  a^  =  a  .  a  •  a  "•  to  r  factors. 

.'.«'"•«'■=(«•«•  a  •••  to  m  factors) (a  •  a  •  a  •••  to  r  factors) 
=  a  •  a  •  a  '  a  '"  to  m  +  r  factors 
=  «""+*■,  by  the  definition  of  exponent. 

Similarly,  a""  •  a*"  •  a^  •••  =  «"'+'■+ p  «.. 

ORAL     EXERCISES 
Multiply : 

1.  a?  .  a\     4.   w}  .  m\      7.    (-  1)^  .  (-  1)5.     lo.    23.23.  22. 

2.  a^  .  aK     5.   x  -  x.  8.    6^  •  6\  11.    7  •  7^  •  73. 

3.  a'  .  a\     6.   23-2^        9.    5  •  5  •  51  12.   3  ■  3^  .  32. 

13.    (-  1)2  .  (-  1)3  .  (_  1)5.       14.    (-  ay  .  (-  ay  •  (-  a). 

355 


356  A   HIGH   SCHOOL  ALGEBRA 

497.   Law  of  Exponents  in  Division. 

IL  —  =  0'"-'',  if  m>r. 

a' 

For    a'^  =  a  ■  a  •  a  ■•'  m  factors, 
and  aj'  —  a  •  a  •  a  •••  r  factors. 

a^_  a  •  a  ••'  m  factors 
'a'"       a  ■  a  '■•  r  factors 

=  a  •  a  '  a  •'•  m  —  r  factors,  canceling  the  r  factors  from  both 

terms 
=  a^-'',  by  definition  of  exponent. 


ORAL    EXERCISES 


Divide 


1.  ^.  5.    t.  9.    ^.  13.    ^. 

a^  a^  6^  X 

2.  2!.  6.    i^^.  10.    ^.  14.    ^. 
a^                      (—a)                       5^  V 

3.  ?L.  7.    i^ll'.  11.    i=^.  15.    M^ 

4.  ?!.  8.    MZ.  12.    5?.  16.    ^. 

22  (c66)  a;  r 

498.   Laws  of  Exponents  for  Powers. 

in.  (a"'y  =  a""'. 

For     (a"*)*"  =  a"^  •  a"^  •  a"^  "•  to  r  factors 

=  (a  •  a  •••  to  m  factors) (a  •  a  •••  to  m  factors)  to  r  such 

parentheses 
=  a  ■  a  •  a  '•-  to  mr  factors 
=  a'"'',  by  definition  of  exponent. 


ORAL  EXERCISES 

Apply  this  law  to : 

1.  (43)2.              4.    (a^y.  7.   (a;2)5.              lo.    [(6)^^ 

2.  (32)5.               5_    (^2)3,  8.    (oc^y.               11.    [(-a)5]2. 

3.  (2^)4.            ■    6.    (a^y.  9.    (2/0'-               ^2.    [(- 8)2]3. 


EXPONENTS  AND  ROOTS  357 

IV.  (aby  =  a"b". 

For  (aby  =(ab)  •  (ab)  •••  to  n  factors 

=  (a  •  a  •••  to  w  factors) (b  ■  b  •"  to  n  factors) 
=  a"?)'*,  by  deiSnition  of  exponent. 

Similarly,  (a&c  •••)*'  =  a'^^^c"  •••. 


ORAL    EXERCISES 

Apply 

this  law  to : 

1.    (8 

.3)^. 

4. 

(aby.             7.    (mny. 

10. 

(x'yy- 

2.    (4. 

.5)^. 

5. 

{cdy.              8.    (xyy. 

11. 

(xyy. 

3.    (2 

.5)3. 

6. 

(abcy.            9.   (aby\ 

12. 

(aa^y. 

V. 

fa\" _ a" 
\b)       b"' 

For 

/ay 

\bj 

f  =  ^.  ^...  to  w  factors 
'       b    b 

a  •  a  •••  to  n  factors 
b  -b^-to  n  factors 

Apply  this  law  to : 


1. 

(W- 

2. 

ay- 

3. 

ar- 

4. 

ay- 

5. 

(-f)'- 

6. 

(-1)^. 

7. 

(!)■■ 

— ,  by  definition  of  exponent. 

5» 


ORAL    EXERCISES 


8.      -    .  12. 


9.    ,|,. 


10.    (  -  )  .  14 


11.    ('^Y  15 


358                        A  HIGH  SCHOOL  ALGEBRA 

499.    Collected  Laws  of  Exponents. 

I.   a*" .  a*-  =  a"'+^  Sec.  496. 

II.   a'"  -T-  a'-  =  a"*-^.  (m  >  r.)          Sec.  497. 

IIL    (a'^)'-  =  a""-.  Sec.  498. 

IV.    (aby  =  a^b\  Sec.  498. 

V.   f'^Y  =  ^.  Sec.  498. 


b     ~b^ 


FRACTIONAL  EXPONENTS 


500.  Hitherto  we  have  spoken  only  of  positive  integers  as 
exponents,  the  exponent  meaning  the  number  of  times  the 
base  is  used  as  a  factor.  This  meaning  does  not  apply  to 
fractional  and  negative  exponents,  because  it  does  not  mean 
anything  to  speak  of  using  a  as  a  factor  |  of  a  time,  or  —  6 
times.  But  it  is  possible  to  find  meanings  for  fractional  and 
negative  exponents  such  that  they  will  conform  to  the  laws  of 
integral  exponents. 

501.  Preparatory. 
Find  the  meaning  of  a^. 

Assuming  that  Law  I  appHes,  a^  >  a^  =  a^"^^  =  «» 
or  (a^)2  =  a. 

That  is,  a^  is  one  of  the  two  equal  factors  of  a, 

or,  a^  =  Va. 

Thus,  the  fractional  exponent  |  means  square  root. 

Similarly,  a^  .  a^  .  a^  =  a^+J"^^  =  a, 

or,  (a^)3  =  a. 

That  is,  a^  is  one  of  the  three  equal  factors  of  a, 

or,  a^  =  Va. 

Thus,  the  fractional  exponent  |  means  cube  root. 


EXPONENTS  AND  ROOTS  359 

WRITTEN    EXERCISES 

Find  similarly  the  meaning  of : 

1.  ai  3.    xK  5.    ni  7.    mi 

2.  a\  4.    6tV.  6.    cK  8.   a^'^, 

i 

502.  The  meaning  ot  a"  is  found  as  follows  : 

Assuming  that  Law  I  applies, 

i         1         i         ^     ^      ^  l  +  Ul+.-.tonterms 

a"  •  a"  •«"■•••  to  w  factors  =  «»*    **    »* 

=  a     " 
=  a^  =  a. 
1  1 

That  is,  a  **  is  one  of  n  equal  factors  of  a,  or  a  "■  =  Va. 

503.  Preparatory. 

Find  the  meaning  of  a^. 

Assuming  that  Law  I  applies,  a^  •  a'  •  a^  =  a^  =  a^,  or  (a'^y  =  a^. 
That  is,  aJ  is  one  of  the  three  equal  factors  of  a^,  or  a^=\/a^. 

P 

504.  The  meaning  of  a'  is  found  as  follows : 
Assuming  that  Law  I  applies, 

t        P         P  H  +  ?!  +  ?^...t0  3term, 

a?  .  a?  .  a?  ...  to  ^  factors  =  a*    2    « 

q.P 

=  a    1  =  aP. 


p 


That  is,  a  3  is  one  of  the  q  equal  factors  of  a^, 

P         q, 

or,  a~'  =  V  «^- 

Similarly,  a^  =  a^     =(Va)p. 

In  words : 

a  with  the  exponent  -  denotes  the  qth  root  of  the  yth  power  of 

a,  or  the  i^th  power  of  the  qth  root  of  a. 

This  definition  applies  when  p  and  q  are  positive  integers.     The  mean- 
ing of  negative  fractional  exponents  is  found  in  Section  513,  p.  366. 
24 


360  A  HIGH  SCHOOL  ALGEBRA 


ORA 

L    EXERCISES 

State  the  meaning  of : 

1.   5'i                 3.   eK 

5.    4l 

7.    8l 

2.    a\                  4.    ai 

6.  bi 

8.    cK 

Find  the  value  of : 

9.    si  10.    lel  11.    25i  12.   32l 


WRITTEN    EXERCISES 

Express  with  fractional  exponents : 

1.  </a\  ^^    ^j32a'b\  20.    Va. 

2.  -v/^.  '''  21.    Va.Vft. 


3.    ^^.       .  12.     ^^^^^' 


22.    \/a"». 


^aVb 


9  62 

-8a^2/6  23.    Va^ 


1    Q 

5.  -y/ab.  ■    "V       5        '  24.    VP". 

6.  </M^.  i^_    ■■"/fi^^  25.    V^-. 


.    V/^'.  ,^  26.    '-V^^. 


'^*    ^'   4    '  15.    Va  +  6. 


•  V- 


25  16.    ^a^-\-b^. 


27.    "Va^ft"*. 


17.    ^P.Va.  2^-    ^«'^5. 

9.  ■^ie'^.  18.  ^:r^.-v/"=^.    29.  V^. 

10.    Vm^^^.         19.    -s/(a  -  by. 


505.  The  definition  of  positive  fractional  exponents  has  been 
found  as  a  consequence  of  the  assumption  that  Law  I  applies 
to  them.  It  can  be  shown  that  the  other  laws  of  Section  499, 
p.  358,  also  apply  to  this  class  of  exponents,  as  thus  defined, 
and  we  shall  so  apply  them,  although  the  proof  is  omitted 
here. 


EXPONENTS  AND  ROOTS  361 

506.    According  to  Law  I  (Sec.  499),  lohen  the  bases  are  the 
same,  the  exponent  of  the  product  is  found  by  adding  the  exponents. 

1  3  1.3  5 

For  example  :  a^  .  a^  =  a"^  ^  =  a^ 

A  general  formula  for  this  statement  is, 

m  p  mq      np  mq+np 

a'*  •  a^  =  a'"'    "*  =  a  "*   . 
The  number  a,  or  the  base,  must  be  the  same  in  all  factors.     When  it 

1  3 

is  not,  as  in  a^  •  6^,  the  product  cannot  be  found  by  adding  the  exponents. 


WRITTEN    EXERCISES 

Find  the  products : 


1. 

a^-ai 

6. 

at.  a*. 

1    .  1 
11.    x^.a^. 

2. 

42 .  4f . 

7. 

m^  •  m^. 

m         p 

3. 

ji.ji 

8. 

ai.ai. 

12.    x""  '  of. 

4. 

a^  •  a\ 

9. 

1           1 

?'"  •  r^. 

13.  p^  'p^  '  p. 

5. 

b^  .  6l 

10. 

1       1 
a**  •  a*". 

14.   a^  '  a^  '  aK 

507.  According  to  Law  II  (Sec.  499),  when  the  bases  are  the 
same,  the  exponent  of  the  quotient  is  found  by  taking  the  difference 
between  the  exponents. 

For  example  :  a^  -=-  a^  =  a^^  =  d^. 

3         i-a         1 
a^  a^  =  a      =  a^. 

A  general  formula  for  this  statement  is, 

m  p  m      p  mq—np 

a"^  -5-  a^  =  a"~^  =  a  "^   , 

—  is  here  supposed  to  be  greater  than  ^,  but  this  restriction  will  be 
71  q 

removed  later. 


362  A  HIGH  SCHOOL   ALGEBRA 


\ 

WRITTEN    EXERCISES 

Find  the  quotients 

; 

« 

1.   a^--ai 

7. 

J^ai 

12. 

m            p 
X""  -^  X'^. 

2.   a-i-d^^. 

8. 

x^^xi 

1              1 

1 

13. 

m«  -=-  m^ 

3.    an- a". 

9. 

ah  -  iab)^. 

6t^6^ 

p 

14. 

4.   a-ira^. 

fl\^      /1\^ 

5.    a^-r-ai 

10. 

15. 

m^  -7-  m*. 

6.    ips  -^  a;^. 

11. 

5*  H-  5i 

16. 

p^  -J-  p^. 

508.  According  to  Law  III  (Sec.  499),  when  an  exponent  is 
applied  to  a  base  having  an  exponent,  the  product  of  the  exponents 
is  the  exponent  of  the  result. 

1  2  •   ^  >  ' 

For  example :  (a^)^  =  o     ^  ■=  a}  =  a. 

A  general  formula  for  this  statement  is, 

m    p  m     p  mp 

(a")«  =  a"' «=  a"^. 


ORAL    EXERCISES 
Simplify  by  applying  Law  III : 

1.    (2^)1            5.    (6*)i              9.    (aj^)l  13.  (a^)i 

12.    (3^)2.            6.    (a^)l            10.    (/)i  14.  (a^)^ 

3.  (3^)3.             7.    (a^)l            11.    (5^)t  15.  (10^)1 

4.  (5*)2.            8.    (a^)l            12.    (3*)i  16.  [(a+6)^]^ 
17.    [(a-6)^]l           18.    [(a2-62)i]l.  19.    ^(x-^yyj^^ 


EXPONENTS  AND  ROOTS  363 

509.  According  to  Law  IV  (Sec.  499),  an  exponent  affecting 
a  product  is  applied  to  each  factor,  and  according  to  Law  V 
(Sec.  499),  ayi  exponent  affecting  a  fraction  is  applied  to  both 
numerator  and  denominator. 


For  example : 


(8  :>fiyH)^  =  %^x^y~H^  =  2  x'^yh^- 

m     p  mp        mp        p 


m- 


(y8)i    yi   y' 


A  general  formula  for  an  exponent  affecting  a  fraction  is, 


WRITTEN   EXERCISES 
Simplify  by  applying  Law  IV: 

1.  (a%^)K  5.    (16  xV)^.  9.    {S6x^y)\ 

2.  {a'bfy,  6.    (a%n^)K  ^o.    f^^Y' 

3.  (a-bny  7.    (27a-6V)i  [     . 

510.  When  the  bases  are  different  and  the  fractional  expo- 
nents are  different,  the  exponents  must  be  made  to  have  a 
common  denominator,  before  any  simplification  is  possible. 

For  example  :  a-&^  =  ah^  =  (a^b"^)^. 

A  general  formula  for  this  statement  is, 

m    p  mq  np_  1 

«"&«  =  a"*  •  6"'  =  (a'"«  •  b^py^. 
This  is  simplifying  by  reducing  exponents  to  the  same  order. 


364  A  HIGH  SCHOOL  ALGEBRA 

WRITTEN   EXERCISES 

Simplify  by  reducing  the  exponents  to  the  same  order : 

1,1  1,3  111 

1.  a2  .  63.  5.  a3  .  6?.  9.  m^-n^'p^. 

2.  a*  .  65  6.  h^  •  ei  10.  m*  •  n^  -^  p^. 

3.  v^  '  bK  7.  5^  •  bK  11.  p^  '  q^  '  r^. 

4.  a^ -b^.  .         8.  a^^6i  12.  x'^-y^-z^. 

511.  It  is  usually  preferable  to  indicate  roots  by  fractional 
exponents  instead  of  by  radical  signs,  since  operations  are  thus 
more  easily  seen. 

COMPARISQN 

By  Kadioals  By  Exponents 

1.  y/aVa  =  v^v^  =  V^^  =  \^.  a2a^  =  J  J  =  at+t  =  «!. 

2.  Vb-^Vb  =  -\/bs^V¥=VWT¥'  =  Vb.  bi-^bi  =  h^^b^  =  b's-l  =  hh 


WRITTEN     EXERCISES 

Simplify  by  use  of  fractional  exponents  as  in  the  examples 
above  : 

1.  2^.5^  9.    V8l^'.  17.  3^.2.7^ 

2.  V^.  10.    V5  .  V75.        18.  3.5^.  2  V3. 

3.  V8.3V2.  11.    (16a^p)k  ^^  f4.9  a'\\ 

4.  2V3.3.VIO.      12.    a/32'^V.  ^^^^'^ 

5.  2^3.3^2.  13.    V36l^.         20.  <^aW2E^. 

6.  5^  .  3  .  5\  14.    V64  m%«.         21.  '\^x'y<^25  yz. 

7.  V7-lli  15.    (49a46«)i  22.  ^/a^^l 

8.  -8V2.I2V3.    16.    ■</2l(^\  23.  V</4^. 


EXPONENTS  AND  ROOTS  365 

24.  -V12-2V3.     32.    V32.V2.       40.    V^sT^. 

25.  aV6.6Va.  33.    (a26^^2)3.       ^^     -^^^^7g^2. 

26.  (V2-V3)2V3.34.    (m^^?)^  3/ ,-==: 

27.  3^(6^-2.5^).    3^-    V6a6.V2a. 

36     r2aaa2^il2     43.    a(a26)^  •  6(«?>V- 

29.  V^«-  3'^-    11^ -11^  •11^-44.   3V9^-3^- 

30.  -{/%~G^^.  38.    "V'm2A/m2.         45.    5^/a^ -2^5/0^. 

31.  (27a;i/2)i  39.    V-^/g^.  46.   3  V8  •  2-v/6  •  3-v/54. 

MEANING   OF  ZERO  AND   NEGATIVE  EXPONENTS 
512.    The  meaning  of  a^  may  be  found  as  follows  • 

Assuming  that  Law  I  holds  for  a°,         w*  -a?  —  a^+o 

=  ««. 

Dividing  by  a^^  aO  =  —  =  1. 

That  is  : 

^ni/  number  {not  zero)  with  the  exponent  zero  equals  1. 

Thus,  60  =  1,  100  ^  1^   (Y-)o  =  1,   [^]*^  =  1. 


ORAL 

EXERCISES 

State  the  value  of : 

1. 

{ah)\ 

5.    (-3)0. 

10.    |.aO. 

15. 

(1)°- 

2. 

.90. 

6.  {\r 

11.    a^ao^ 

16. 

.9  - 100«. 

3. 

100°. 

7.    (-2V)°. 

12.    a"*  .  50. 

17. 

91- .  50. 

4. 

(sj- 

8.  32.30. 

9.  32.50. 

13.  w'  -V-  aO. 

14.  43 --40. 

18. 
19. 

30 .  270. 
(23  .  32)0. 

366  A  HIGH   SCHOOL   ALGEBRA 

513.    The  meaning  of  the  negative  exponent  may  be  found 
as  follows : 

Assuming  that  Law  I  holds  for  negative  exponents, 
5-3  .  5+3  _  5-3+3  -  50  _  1. 

That  is,  5-3  is  a  multiplier  such  that  its  product  with  5+3  is  1.     But  if 
the  product  of  two  numbers  is  1,  one  is  the  reciprocal  of  the  other. 

Therefore,  5-3  is  the  reciprocal  of  5^  which  is  — • 

53 

Expressed  in  general  terms : 

a~^ .  a"  =  «-"+" 
=  0^  =  1. 

In  words : 

a"°  means  — ,  for  all  values  of  n,  positive  or  negative,  integral 
or  fractional. 

ORAL    EXERCISES 
Find  similarly  the  meaning  of : 

1.  4-3.  3.    (|)-l 

2.  2-\  4.    (-5) 
State  the  value  of : 

9.   4-1  13.  16"i 

10.  8"i  14.  16"^. 

11.  S-\  15.  .125-i 

12.  (.2«)i  16.  .125-1 

WRITTEN    EXERCISES 

Perform  the  operations  indicated,  admitting  negative  expo- 
nents to  the  results : 

1.  (42  .  5')-K  3.    2*  .  2-1  5.   ^  '  •^\ 

2.  214 .  2-6.  4.    2"^  .  2~K  6.    10-5 .  iQ2  .  i()o_ 


-6 


5. 

a-^ 

7. 

©-«• 

6. 

a-K 

8. 

(-!)"*• 

17. 

32-i 

21. 

a-^ 

18. 

27-1 

22. 

(«»)-». 

19. 

.36-i 

23. 

«-l 

20. 

64-1 

24. 

a^. 

EXPONENTS   AND  ROOTS  367 

7.  2-t.2f.  9.    3^.3-^.40.  11.    10^.10-^ 

8.  27"^  .  9i  10.    {x-^y.  12.    a"  •  ai 

13.  ■v'lS^*  •  ^18^^  18.   ^(ora"^-^-^^). 

14.  i/W'^Vw.  "",     _, 

19.  a^  •  a  ^(or  a^  ^). 

15.  (10-'»  .  10-3)1  1 

20.  -^(otJ-^-^K 

16.  (10-3.10^.  a-^ 

17.  loK  loK  loK  (4)0.  21.    (a-V*(a^"y- 

USE  OF  ZERO,  NEGATIVE,  AND  FRACTIONAL  EXPONENTS 

514.  We  have  defined  zero  and  negative  exponents  so  that 
Law  I  holds  for  them.  It  can  be  shown  that  the  other  four 
laws  hold  for  these  exponents  as  defined,  but  the  laws  will  be 
applied  here  without  proof. 

515.  The  relation  a""  .  a"  =  a°  =  1  can  be  used  to  change  the 
form  of  expressions. 

I.  To  free  an  expression  from  a  negative  exponent,  multiply 
both  numerator  and  denominator  by  a  factor  that  tvill  so  combine 
with  the  factor  having  the  negative  exponent  as  to  produce  unity. 
If  more  than  one  negative  exponent  is  involved,  apply  the  process 
for  each. 

For  example  :  7-3  .  2*  =  ^-^ =  — . 

^  73  73 

gg  ^  &5 .  q,3  ^  55^3  ^  ^^^^        • 
5-0      55 .  5-5         1 
x-^  _tH'^- x-^  _  t^ 
t-^     t^x^  •  t-^  ~  x^' 


WRITTEN    EXERCISES 

Write  equivalent  expressions  without  negative  exponents 


2-3  X-*  6-V 


368  A  HIGH   SCHOOL   ALGEBRA 


0' 

-t3 

X"' 

5. 

_ 

7. 

b- 

-3 

a~- 

8- 

-2  . 

4-1 

a~'' 

«. 



8. 

5- 

-3 

b' 

9.   ^.  11.   5l!_ 

10.   a-16-2.         12.   ^^ 


a'^'x' 


II.  To  free  an  expression  from  a  fractional  form,  multiply 
both  numerator  and  denominator  by  a  factor  that,  in  combination 
with  the  denominator,  will  produce  unity.  If  more  than  one  such 
form  is  involved,  apply  the  process  for  each. 

For  example : 

62        6-262 

also,  A+^5!_  =  iLA+  ^'-^w 


4"^     x^y~^     4^  •  4~^     x'^y^x^y'^ 
45  .  2  +        a;- V«^ 


(x-^x'){y^y-^) 
=  2  .  45  +  a3x-2?/3. 

WRITTEN    EXERCISES 

Free  from  fractional  forms  : 

1.  ?_.  3.  JL,  5.  i  +  ?-.  7.   — -h— . 
52                       6a:3                         52      23  y--'      x-^ 

2.  J-.  4     ^.  6.    ^'  +  ^.  8.    ^  +  ^. 
a'b^              *•    r^                        /      x^  5      6-1 

III.  To  transfer  any  specified  factor  from  the  numerator  into 
the  denominator,  or  vice  versa,  multiply  the  numerator  and  de- 
nominator by  a  factor  that,  in  combination  with  the  factor  to  be 
transferred,  will  produce  unity. 

EXAMPLES 

1.    Transferring  the  factors  of  the  denominator  to  the  numerator : 


a;*y"3     x'^y^x^y 


-3 


-4?/3x'^  =  x^y^ 


2.   Transferring  the  literal  factors  of  the  numerator  to  the  denomi- 
nator : 

5  aW  ^  5  a-^b-'^a^^  ^      5 
4  a45-3     4  a-%-H^h-^     4  ah'^ ' 


EXPONENTS  AND  ROOTS  369 

WRITTEN    EXERCISES 
In  the  following  expressions : 
(a)  Transfer  all  literal  factors  to  the  numerator. 
(6)  Transfer  all  literal  factors  to  the  denominator. 

2.    ^.  4.    ^^!5!^.      ^      a-^6-^ct         ,     6a.-V 

516.    The  laws  of  exponents  enable  us  to  perform  operations 
with  polynomials  containing  fractional  and  negative  exponents. 

Thus  :  (a^  +  &^)2  =  (a^y  +  2  ah^  +  (&^)2 

WRITTEN    EXERCISES 

Perform  the  indicated  operations,  admitting  negative  expo- 
nents to  the  results : 


1. 

(a^-bfy. 

8. 

a' -of 

5 

2. 

(4  aM  +  23)2. 

9. 

{x^-y-y. 

3. 

10. 

(a2n63r_-L)2, 

4. 

{a-  -  3  feO'. 

ap  •  6' 

5. 

(^-f_2/-|)(^- 

Uy-h 

11. 

ap+i  .  6«-i 

6. 

(..-  + !)(.;- 

-!)• 

12. 

(a"  +  r)2. 

7. 

(x^  +  3){x'^-^ 

5). 

13. 

(a^7>^+a;-^) 

Express  as  a  product  of  two  factors : 

14.  x^—m-\  18.    y-'^—x'^^, 

15.  a^  —  2  a%^^  +  x^.  19.    cC  —  4. 

16.  a^n  _|_  2  an^n  _,_  ^2n^  20.    1  +  8  a;"^  + 16  a;-^ 

17.  ic4n  _  4  ^«2^2h  _|_  4  ^4n^  21.     OJ^^  _|_  g  ^.6^-1  _j_  Q  ^/-T. 


370  A   HIGH   SCHOOL  ALGEBRA 

EVOLUTION 

Square  Root  of  Binomials  of  the  Form  a  +  V6 

517.  Binomials  of  the  form  a  +  V5  can  often  be  put  into  the 
form  x-^y  -{-  2^xy,  or  (V^  +  V^)^  and  hence  the  square  root, 
Va^+V^/,  of  the  binomial  can  be  written  at  once. 

EXAMPLES 
1.   Find  the  square  root  of  4  +  2V3. 

4  +  2  VS  =  3  +  1+  2\/3TT. 
Hence,  a;  +  ?/  =  3  +  1  and  xy  =  3  •  1,  from  which  cc  =  3  and  y  =  1. 


•••  V4+2\/3=±(V3+VI)  =  ±(V3+l). 

The  coefficient  of  the  radical  must  be  made  2  in  order  to  apply  the 
formula  x  +  y  +  2\/xy. 

2.  Find  the  square  root  of  3  —  V8. 

3  -  VS  =  3  -  VTTI  =  3 -2  V2. 
.*.  X  +  ?/  =  3,  and  xy  =  2.     .*.  x  =  2,  ?/  =  1  by  inspection. 

...  V3-V8=±  (V2_vT)  =  ±(\/2-l). 

3.  Find  the  square  root  of  7  -+  4V3. 

7  +  4V3  =  7  +  2V4.3;  x  +  y  =  1,xy=\2]  .-.  x  =  4,  y  =  3. 

.-.  Vt  +  4 V3  =  ± ( V4  +  V3)  =  ± (2  +  VS). 

The  square  root  as  a  whole  may  be  taken  positively  or  negatively,  as  in 
the  case  of  rational  roots. 

The  solution  of  these  problems  depends  upon  finding  two  numbers 
whose  sum  and  product  are  given.  This  can  sometimes  be  done  by 
inspection,  but  the  general  problem  is  one  of  simultaneous  equations. 


WRITTEN    EXERCISES 
Find  the  square  root  of  : 

1.  11  -f  6V2.  4.  41  -  24 V2.  7.  17  +- 12 V2. 

2.  8-2V15.  6.  2i-V5.  8.  |V5  +  3f 

3.  49  -  12 VlO.  6.  2\  -  |V3.  9.  m  -  24 V5. 


EXPONENTS  AND   ROOTS  371 


Cube  Root  of  Arithmetical  Numbers 

518.  Pointing  off  into  Periods.  Since  lO^  =  1000,  we  know  that  the 
cube  root  of  any  number  greater  than  1  but  less  than  1000  is  less  than  10. 
Its  integral  part  consists  of  one  figure. 

Since  100^  =  1,000,000,  we  know  that  the  cube  root  of  any  number 
greater  than  1000  but  less  than  1,000,000  is  greater  than  10  but  less  than 
100.  That  is,  if  the  given  number  has  from  4  to  6  digits  in  its  integral 
part,  its  cube  root  will  have  2  digits  in  its  integral  part.  If  larger  num- 
bers are  given,  the  above  reasoning  can  be  repeated  for  1000^,  etc.,  show- 
ing that  in  all  cases  if  the  number  be  pointed  off  into  periods  of  3  digits 
each  (or  possibly  fewer  in  the  left  period),  then  each  period  will  corre- 
spond to  a  digit  of  the  root. 

519.  The  cube  root  of  the  left  period  can  be  found  approxi- 
mately by  inspection,  and  the  number  so  found,  with  a  zero 
annexed  for  each  other  period,  will  be  an  approximate  value 
for  the  root. 

EXAMPLE 

Find  -v/481890304. 

Pointing  off  as  above  : 

481'890'304. 

By  trial  we  find  7^  =  343,  and  8^  =  612.  Hence  700  is  an  approximate 
value  for  the  root.  It  may  be  verified  that  700^  is  less  than  the  given 
number,  and  that  800^  is  more  than  the  given  number.  That  is,  the  hun- 
dreds' figure  of  the  root  is  7. 


WRITTEN     EXERCISES 

Find,  as  above,  the  first  figure  of : 


1.  V493039.  3.    V57066625.  5.    V1345572864. 

2.  ^2924207.  4.    ■v/254840104.  6.    ^62287505344. 

520,  When  once  an  approximate  value  a  has  been  found  for 
the  root,  an  approximate  value  for  the  remainder,  r,  of  the  root 
can  be  found  by  means  of  the  formula: 

(a  -f  r)  3  =  a^  -f  3  a^r  +  3  ar^  -f  r^. 


372 


A  HIGH   SCHOOL   ALGEBRA 


For  example 


In  V238328  we  find  as  above  that  a  =  60. 

Then,  a^  +  Sa^r  +  3  ar^  +  r^  =:  238'328,  the  whole  cube 

Subtractmg  a^ =  216000,  the  cube  of  the  part  found, 

3  a^r  +  Saf^+r^=   22 '328,  the  first  remainder. 

Since  something  must  be  added  to  3  aV  to  make  it  equal  to  22,328, 
3  a'^r  is  less  than  22'328, 


or 


r  is  less  than 


22^328 
3a2 


Consequently,  the  first  figure  of  this  quotient  will  either  be  the  first 
figure  of  r  or  greater  than  it.  In  this  instance  3  a^  is  10,800,  hence  the 
first  figure  of  the  quotient  is  2. 

Trying  2  as  r,  we  have  to  calculate  3  a^r  +  3  ar^  +  r^.  This  is  most 
conveniently  done  by  using  the  form  (3  a^  +  3  ar  +  r^)r. 


We  have  already                               S  a'^  =  10800. 
We  find  :                                             3  ar  =      360 

r2=         4 

Adding :                          3  «2  _^  3  ar  +  r^  =  1 
Then,                        r(3  a^  +  S  ar  +  r^)  =  2. 

The  calculation  should  be  arranged  thus : 

L164 
2328 

6     2 

238'328 
216  000 

Trial  divisor :                             S  a^  =  10800 
3  ar  =  3  .  2  .  60  =     360 

r2  =  22  =         4 

22328 

Complete  divisor :                                11164 

22328  or  2  X  11164 

If  the  root  consists  of  more  than  two  figures,  the  above  work  is  repeated, 
using  the  part  of  the  root  already  found  as  a. 

If  it  should  happen  that  the  product  of  r  and  the  complete  divisor  is 
larger  than  the  remainder  of  the  number,  try  the  next  smaller  digit  for  r. 

When  necessary,  periods  are  pointed  off  to  the  right  of  the  decimal 
point. 


Find 


2.    -^148877. 


WRITTEN    EXERCISES 


3.    V2803221. 


4.    ^16387064. 


5.    V55306341. 


6.    V143055667 


EXPONENTS  AND   ROOTS  373 

Find  these  roots  to  one  decimal  place : 


7.    V637.  8.    V3485.  9.    V263488. 

Find  these  roots  to  two  decimal  places : 
10.   ^5.  11.    ^17  12.    i/m,  13.    ^/4?^63. 

Cube  Root  of  Polynomials 

521.   The  cube  roots  of  polynomials  may  be  extracted  in  a 
similar  manner ; 

EXAMPLE 

Extract  the  cube  root  of  27  cc^  —  27  x^y  +  9  xy^  —  y^. 

Eoot  Sx  —  y 


a3  +  3  a2r  +  3  ar2  +  r3  =  Power      27  cc^  -27  x^y  +  9xy^-  y^ 

a  =  Sx  a»=(Sx)^  =  27x^ 

Trial  divisor  =  27  x^    .•.r=—y  —27  x^y  -\-9 xy"^  -  y^ 

Complete  divisor  =  [3(3  ic)2  -  3(3  a;)?/  +  y2]  -  27  x'^y  +  9  a;y2  -  y^ 

Explanation. 

1.  Arrange  the  expression  in  the  order  of  the  pov^ers  of  some  letter, 
as  x. 

2.  Take  the  cube  root  of  the  first  term  of  the  power  for  the  first  term 
of  the  root,  as  3  x. 

3.  Divide  the  second  term  of  the  power  by  3  times  the  square  of  the 
first  term  of  the  root.     The  result  is  the  second  term  of  the  root,  as 

-V' 

4.  If  a  denote  the  approximate  value  of  the  root  already  found  (3  x  in 
the  above  instance),  and  r  the  value  to  be  used  as  the  next  term  (—  y  in 
the  above  instance),  form  the  complete  divisor  Z  o?'  ■\- Z  ar  ■\-  r^,  multiply 
it  by  r,  and  subtract. 

5.  If  there  is  a  further  remainder,  proceed  as  before,  using  the  entire 
part  of  the  root  already  found  as  a. 

WRITTEN     EXERCISES 

Extract  the  cube  root  of : 

1.  8a^  +  12  ir2  + 6  a; +  1.  4.  7? —  'ix^y -^'d  xy'^  -  y^. 

2.  27-27  a;+9»2_a^.  5.  Sa;^  _  12  a^  +  6a;2-l. 

3.  8a363_12a262  +  6a6-l.  6.  x^"^  -  Z  y^m  ^  Z  oi^w?  -  m? . 


874  A   HIGH   SCHOOL   ALGEBRA 

7.  _a6-6a5  +  40a3_96a  +  64. 

8.  54  ar^/ -  36  a^y  +  8  iK^  -  27  2/3^ 

10.  -64  +  96a;-40a^  +  6a;5  +  a!6. 

11.  27f-27f  +  63y-S7  +  ^-^-{-^. 

y     y"     2/' 

12.  8x3-12x  +  --i. 

13.  a^  _  63  4-  c3  +  3  a62  +  3  ac^  _  3  ftc^  +  3  a^c  +  3  Wc 

—  3  a^^  —  6  a6c. 

14.  27a^-27a;+---. 

X      or 

15.  64(a6)3^-48(a6)2^  +  12(a6)^-l. 

16.  30  a-^  +  8  a-3  +  8  a^  +  30  a  -  12  a2  -  25  -  12  a-^. 

17.  a«-20a3-6a  +  15a4-6a^4-15a2  +  l. 

18.  a^-12a^^  +  54a^-112+  — --  +  -. 

19.  y?^  —  y^  —  z^'  —  3  x^py^  4-  3  a:V*  —  3  x^^z'  +  3  af ;22r  _  3  y-2^^r 

-  3  y^z^'  4-  6  a;^2/'2'*. 

20.  21:x?  +  f--  +  9xy''  +  ^^-21x'y-21--n^ 

z^  z^  z  z 


REVIEW 

WRITTEN     EXERCISES 

Express  with  positive  exponents : 

1.    m~^n8.        2.   ^x'^y-h.        3.    3  a-^6l        4.    17  ic"^2/'^^~^- 

Transfer  all   literal   factors   from  the   denominator   to  the 
numerator : 

x^                        ab                            1                           5 
5.     — -.  6.    — .  7.    -.  8.    -.  . 


EXPONENTS  AND  ROOTS 


375 


Multiply : 
9.    (2  +  V^+l)2. 

10.  V5  .  V6. 

11.  5-y/rnF^ '  2  m-\ 

Kemove  the  parentheses : 
15.    (a-iy.  18.    (Ij)' 


12.  p  .  p 


■I 


13.  (a-i-6-i)(a-^-&-^> 

14.  (x'-l)(x^i.l). 


1  \^ 


21 


^x  « 


16.    (^ci-i)l 


17.    (-1-) 

\</py 


I      5n\    3 


22. 
23. 


(.- 


Simplify  as  far  as  possible,  admitting  negative  exponents  to 
the  results : 


24.  (Sa'bh-^)K 

25.  (16ahc-^)^. 

26.  (5  00^3^2)^^ 

27.  (A/4«2)i. 

28.  (64:X-^y^)~K 

29.  (- 250  a^2/"^^-3)l 

30.  (- 2^3  a-^y-h^yi 


37.  (3a-2--6-2)-5. 

38.  (aP-9)p+«  .  a^V. 


39. 


Va 


0352 


■\/aW    ^/a^b^ 


31.   ^"Vic^^-^^ 


40.  x~^  •  ic^  •  a;2  •  Vic. 

41.  — :::  •  —  • . 

42.  [Ka'-&-')-M"']'- 

43.  [(a-i)^]-i2. 


32.  ->!/i»"2P2/-p^2-i'. 

33.  Va -"*&*'". 

34.  3ahh-^-2ahh. 

35.  10  a5-^c^ -- 2  a-ift^^l 

36.  a"'b~^''c~^  -i-  ab^z-^. 

25 


44.  |Va6-V^|^ 

45.  K«"'^')'^r^- 


46.    VaV^. 

47.  [(x^yy-a-^Kx-yyi 

48.    -a^ft-V^d^.-a-^Z^^c^^-l 


376  A  HIGH  SCHOOL  ALGEBRA 

49.  a^x-S^f.a^xy,  ^^    ^j^^WW^^, 

50.  3  a;~^5  a;^  .  10  a;~i  52.    ]x~^y(xy-yi{x-'^y)^K 


53. 


— y.  _l_  i  f     n±l  n_l\  n-1 

2i/ +  a?      2  54.    l^a?"-i -- a.'"+i )  2- . 


3^ 

-G^ 

a^  +  7 
56. 

y) 

55. 

[(aP+«)p-3 

V(2^  +  2- 
4(2^- 

-2" 

-4 

5+v'21 
5-V21 

57.  Add :  J  V45  +  4  V|  -  Vl25. 

58.  Express  the  product  Va'^vV  as  a  single  surd. 

69.   Divide   2  a;^-^  —  5  x^y-"^  +  7  x^~'^  —  5  a.-^  +  2  a;^?/  by 

Find  equivalent  expressions  with  rational  divisors : 


60.   3V2a-^2V3  6.  ^^    a^+Vs 


61.  6  Va2  --  Va5.  a;  -  Va;^  -  1 

62.  V40  a?y  -^  xV5^.  jq.         ^ 

63.  x-\/y -r- y-^x. 


Va  —  V6 


V  a;  +  Va;  +  2/ 

65.  _3V|-f-/^V3.  ^ 

66.  6^/54^- 2 V2ar^.  ^^-    v^M=^-&' 
67    a2V48a?^--2a6</3a62. 
68.    4  aaj  -^  Vaa;. 


73. 


74.    4=.  75.  ^^-^g, 

a  —  Va^  —  aj2  p  4-  Vg 


77.    V«  +  &+Va      ^^  ^g 


3Va;- 

-3H-Va:  +  3 

3Va;- 

7fi 

-3-Vaj4-3 
1 

2  4-v^-V2 

Va;2_^l4-Va.'2-1 

Va  +  ^  -  Va  -  6  ^x"'  +  1  -  Va;2  -  1 


EXPONENTS  AND   ROOTS  377 

Solve : 

79.    3x+^x'-2x+5  =  l.     80.    ^^±^- ^ 


b  a?-2ah  +  h'^ 

2  ah 


81.  — ^  +  V(a  -\-hy-\-ah-x  =  a-{-h+      ,   ^ 

82.  V^(2a  — 6H-Vfl?)  =  3a2  — a6. 

Extract  the  square  root  of : 

83.  7-2VI0.  84.   a  +  26  4-V8a6. 

Extract  the  cube  root  of : 

85.    8aj3-12aj2^^  +  6a^^-2/. 

SUMMARY 

The   following   questions    summarize   the   definitions    and 

processes  treated  in  this  chapter : 

p 

1.  Explain  the  meaning  of  the  exponent  in  a'.       Sec.  504. 

2.  Explain  the  meaning  of  the  exponent  zero.         Sec.  512. 

3.  Explain  the  meaning  of  the  exponent  in  a"".      Sec.  513. 

4.  State  the  Law  of  Exponents  for  multiplication  ;  for  divi- 
sion ;  for  involution.                                                Sees.  496-499. 

5.  How  are  expressions  multiplied  whose  bases  are  differ- 
ent and  whose  exponents  are  different?  Sees.  510,  514. 

6.  How  may  an  expression  with  negative   exponents  be 
changed  to  an  equal  one  with  positive  exponents  ?  Sec.  515,  L 

7.  How  free  an  expression  from  the  fractional  form  by  the 
use  of  exponents  ?  Sec.  515,  II. 

8.  How  may  a  factor  be  transferred  from  one  term  of  a 
fraction  to  the  other  by  use  of  an  exponent  ?         Sec.  515,  III. 

9.  How  may  the  square  root  of  a  binomial  surd  be  found 
by  inspection  ?  Sec.  517. 

10.   What  formula  may  be  used  to  extract  the  cube  root  of  a 
number  or  of  an  algebraic  expression  ?  Sec.  520. 


378 


A  HIGH  SCHOOL  ALGEBRA 


HISTORICAL  NOTE 

The  use  of  exponents  to  denote  powers  and  roots  seem  so  essential  to 
our  present-day  work  in  algebra  that  it  is  difficult  to  imagine  how  algebra 
could  exist  without  this  notation  ;  but  their  general  use  is  comparatively- 
recent.  Vieta  (1591),  the  founder  of  modern  algebra,  had  no  knowledge 
of  them.  In  place  of  our  notation  he  used  a  system  much  like  that  of 
Diophantos.    Thus,  for  the  expression 


Vieta  wrote 


a:3  _  2  cc2  +  5  x  -  3  =  7, 
1  (7-2§  +  5/Y-3aequ.7, 


in  which  O,  Q,  and  iV  are  the  first  letters  of  the  Latin  words  meaning 
"  cube,"  "  square,"  and  "  number."  By  number  is  meant  the  unknown 
quantity,  and  by  C  and  Q  are  meant  the  square  and  cube  of  this  number. 
In  the  sixteenth  century  Stifel  used  integral  exponents,  and  Stevin  in- 
vented fractional  exponents,  if  we  pass  over  the  beginnings  made  by 
Oresme  (1382),  but  it  remained  for  Wallis  and  Newton,  in  the  seventeenth 
century,  to  popularize  these  improvements. 

John  Wallis,  born  in  1616,  was  second  only  to  Newton  among  English 
mathematicians  of  the  seventeenth  century,  and  became  a  professor  of 

geometry  at  Oxford  College  at 
the  age  of  33.  He  was  a 
charter  member  of  the  famous 
Royal  Society  of  Great  Britain, 
founded  in  1663. 

Exponents,  like  the  bino- 
mial theorem  and  many  other 
principles  of  algebra,  were 
standardized  and  made  popu- 
lar through  their  application 
to  concrete  problems,  for 
Wallis,  in  seeking  the  areas 
inclosed  by  various  curves, 
used  the  series  of  powers,  a^, 
x2,  x^,  x^,  a;"i,  x'^,  x-^  in  its 
present  meaning,  thus  placing 
upon  these  symbols  his  stamp 
of  authority.  In  this  way  our 
present  definitions  of  negative 
exponents  and  of  the  zero 
exponent  were  established. 
The  fact  that  these  interpretations  satisfy  the  fundamental  laws  of 
algebra  was  shown  by  Peacock  and  Hamilton,  as  noted  (p.  326). 


John  Wallis 


CHAPTER  XXVII 

LOGARITHMS 

MEANING  AND   USE   OF  LOGARITHMS 

522.  Use  of  Exponents  in  Computation.  By  applying  the 
laws  of  exponents  certain  mathematical  operations  may  be 
performed  by  means  of  simpler  ones.  The  following  table  of 
powers  of  2  may  be  used  in  illustrating  some  of  these  simpli- 
fications : 


1  =  20 

32  =  25 

1024  =  210 

32768  =  215 

2  =  21 

64  =  26 

2048  =  211 

65536  -  216 

4  =  22 

128  =  27 

4096  =  212 

131072  =  217 

8  =  28 

256  =  28 

8192  =  213 

262144  =  218 

16  =  2* 

512  =  29 

16384  =  21* 

524288  =  219 

523.   Application  of  Law  I,  Section  496,  p.  355. 

EXAMPLES 

1.  Find:  8-32. 

From  the  table,                              8  =  2^,  (i) 

and                                                32  =  25.  {2) 

Then,                                       8  .  32  =  23  .  25  =  28,  (^) 

and,  according  to  the  table,         28  =  256.  ,                                  (.4) 

2.  Find :  2048  •  64. 

From  the  table.                      2048  =  211,  _  (^) 

and                                                 64  =  26.  {2) 

Then,     •                     2048  .  64  =  211  .  26  =  2",  (5) 

and,  according  to  the  table,       217  —  131072.  {4) 

Thus,  the  process  is  simply  one  of  inspection.  In  the  above  example 
we  merely  added  11  and  6  and  looked  in  the  table  for  the  number  opposite 
to  217. 

379 


380  A  HIGH   SCHOOL  AI^GEBRA 

ORAL    EXERCISES 

State  the  following  products  by  reference  to  the  table : 

1.  16  .  256.  5.   32  •  32.  9.   128  •  512. 

2.  32  .  128.  6.   64  •  64.  10.   128  •  1024. 

3.  64  .  512.  7.   32  •  2048.  11.   8  •  16384. 

4.  8- .  2048.  8.    16  •  4096.  12.   32  •  4096. 

524.   Application  of  Law  TI,  Section  497,  p.  356. 

EXAMPLES 

1.    Find:   -— -• 
32 

From  the  table,  256  =  2^,  (i) 

and  32  =  25.  C^) 

Hence,  ^  =  |  =  28-5  =  23,  (5) 

and,  according  to  the  table,      2^  =  8.  (^) 

65536 


2.   Find: 


2048 


A     .  65536     216 

As  above,  -iz^rrrr  =  ;;rr;  =  25  =  82. 

2048       211 


ORAL     EXERCISES 


By  use  of  the  table  determine  the  value  of  the  following : 
1024  32768  32  .  2048 

'     128  '  '     1024  '  '         512      ' 

8192  *  64  .  512  128  .  131072 

64   *  '    16  .  128'  *    64  .  8  -8192* 

525.   Application  of  Law  III,  Section  498,  p.  356. 

EXAMPLES 
1.   Find:  161 

By  the  table,  16  =  2*.  (i) 

Hence,  163  =(24)3  =  212.  (^2) 

and,  according  to  the  table,  212  =  4096.  (5) 


LOGARITHMS  381 


2.  Find:  V1024. 

As  above,     1024  =  (1024)  ^  =  (210)  i  ==  2^  =  32,  according  to  the  table. 

3.  Find:  ^32768. 

As  above.  y/Wm  =  (215)  i  ^  £3  =  8. 

ORAL   EXERCISES 
By  use  of  the  table  find  the  value  of : 

1.  321  4.   642.  7,    V8192.  10.   512^ 

2.  3^  5.   2562.  8.    -v/4096.  11.    v'32768. 

3.  325.  6,   ie4_  9^    a/65536.  12.    VTMi. 

526.  The  examples  and  exercises  above  show  that  the  laws 
of  exponents  furnish  a  powerful  and  remarkably  easy  way  of 
making  certain  computations. 

In  the  above  illustrations  we  have  used  a  table  based  on  the  number  2, 
and  have  limited  the  table  to  integral  exponents  ;  but  for  practical  pur- 
poses a  table  based  on  10  is  used  and  is  made  to  include  fractional 
exponents. 

For  example  : 

1.  It  is  known  that  approximately, 

2  =  lOT^  or  10-3  (more  accurately  10-30i). 
From  this  we  can  express  20  as  a  power  of  10,  for 

20  =  10  .  2  =  101 .  ]0-30i  =  101-301, 
Similarly,  200  =  10  •  20  =  lOi  •  lOi-soi  =  I0230i, 

and  2000  =  10  •  200  =  lOi  •  102-30i  =  I03.30i. 

2.  It  is  known  that  approximately  763  =  10283. 

Then  7630  =  10  •  763  =  lOi  •  102-88  =  103.88, 

and  76300  =  100  •  763  =  10^  .  102-88  =  lO^-ss. 

Similarly,  76.3=  —  =  ^^  =  102-88- 1  =  iQi-ss 

^'  10        101 

and  7.63  =  —  =  i^  =  102-88-2  =  100.88. 

100       102 


382  A  HIGH  SCHOOL  ALGEBRA 

WRITTEN     EXERCISES 
Given  48  =  10^-^;  express  as  a  power  of  10 : 
1.   480.  2.   4800.  3.   48,000.  4.   4.8. 

Given  649  =  lO^-^^ ;  express  as  a  power  of  10 : 

5.  6490.  7.   649,000.  9.    6.49. 

6.  64,900.  8.   64.9.  10.   649,000,000. 

Given  300  =  lO^"*^ ;  express  as  a  power  of  10  : 

11.  3.  13.   3000.  15.   300,000. 

12.  30.  14.   30,000.  16.   3,000,000. 

527.   The  use  of  the  base  10  has  several  advantages. 

I.  The  exponents  of  numbers  not  in  the  table  can  readily 
be  found  by  means  of  the  table. 

To  make  this  clear,  let  us  suppose  that  a  certain  table  expresses  al\ 
integers  from  100  to  999  as  powers  of  10 ;  then  30,  although  not  in  this 
table,  can  be  expressed  as  a  power  of  10  by  reference  to  the  table. 

For,  30  =  — — ,  and  since  300  is  in  the  supposed  table  we  may  find 
by  reference  to  the  table  that  300  =  102-47,  and  hence,  30  =  ^-^  =  lOi-^^. 

Similarly,    3.76    is    not    in    the    supposed    table,    but    376    is    and 

376      .376 
3.76  =  — —  =  '— -^  .    Therefore  it  is  necessary  only  to  subtract  2  from 

the  power  of  10  found  for  376  in  order  to  find  the  power  of  10  equal  to 
3.76. 

Similarly,  4680  is  not  in  the  table,  but  468  is,  and  4680  =  468  •  lOi. 
Therefore  it  is  necessary  only  to  add  1  to  the  power  of  10  found  for  468 
in  order  to  find  the  power  of  10  equal  to  4680. 

Such  a  table  would  not  enable  us  to  express  in  powers  of  10  numbers 
like  4683,  46.83,  and  356,900,  but  only  numbers  of  3  or  fewer  digits,  which 
may  be  followed  by  any  number  of  zeros. 

Similar  conditions  would  apply  to  a  table  of  powers  for  numbers  from 
1000  to  9999,  from  10,000  to  99,999,  and  so  on. 

II.  The  integral  part  of  the  exponent  can  be  written  with- 
out reference  to  a  table. 


LOGARITHMS  383 

For  example : 

1.  879  is  greater  than  100,  which  is  the  second  power  of  10,  and  less 
than  1000,  or  the  third  power  of  10.  That  is,  879  is  greater  than  10^  but 
less  than  10^.  Therefore  the  exponent  of  the  power  of  10  which  equals 
879  is  2.  4-  a  decimal. 

2.  Similarly,  S7.9  lies  between  10  and  100,  or  between  lOi  and  lO^, 
hence  the  exponent  of  the  power  of  10  that  is  equal  to  87.9  is  1.  +  a 
decimal. 

ORAL    EXERCISES 

State  the  integral  part  of  the  exponent  of  the  power  of  10 
equal  to  each  of  the  following : 

1.  35.  4.   25.  7.    25.5. 

2.  350.  5.    2500.  8.   365.5. 

3.  36.5.  6.   36,500.  9.    17.65. 

III.  If  two  numbers  have  the  same  sequence  of  digits  but 
differ  in  the  position  of  the  decimal  point,  the  exponents  of 
the  powers  of  10  which  they  equal  have  the  same  decimal 
part. 

For  example : 

Given  that  274.3  =  102-«, 

we  have  27.43  =  ?^  =  ^  =  101-43, 

also  2743  =  10  •  274.3  =  lOi .  102-43  =  i03.43^ 

also  274,300  =  1000  .  274.3  =  103  .  102.43  =  105.43. 

In  each  instance  the  decimal  part  of  the  exponent  is  the  same.  It  is 
evident  that  this  will  be  the  case  in  all  similar  instances,  for  shifting  the 
decimal  point  is  equivalent  to  multiplying  or  dividing  repeatedly  by  10, 
which  is  equivalent  to  changing  the  integral  part  of  the  exponent  by  add- 
ing or  subtracting  an  integer. 

ORAL    EXERCISES 

Given  647  =  10^-^^,  state  the  decimal  part  of  the  exponent  of 
the  power  of  10  that  equals : 

1.  64.7.                            3.   6470.  5.  647,000. 

2.  6.47.                            4.    64,700.  6.  6,470,000. 


384  A  HIGH   SCHOOL   ALGEBRA 

Given  568.1  =  10^''^,  state  the  decimal  part  of  the  exponent 
of  the  power  of  10  that  equals  : 

7.  56.81.  9.   5681.  11.   568,100. 

8.  5.681.  10.   56,810.  12.    56,810,000. 

528.  Logarithms.  Exponents  when  used  in  this  way  for 
computation  are  called  logarithms,  abbreviated  log. 

529.  The  number  to  which  the  exponents  are  applied  is 
called  the  base. 

For  the  purposes  of  computation  the  base  used  is  10. 

According  to  the  above  definition  the  equation  30  =  IQi-^  may  be 
written  log  30  =  1.48,  which  is  read  "  The  logarithm  of  .30  is  1.48. "  These 
equations  mean  the  same  thing  ;  namely,  that  1.48  is  (approximately)  the 
power  of  10  that  equals  30. 

WRITTEN    EXERCISES 

Write  the  following  in  the  notation  of  logarithms : 

1.  700=102**.  3.   6  =  10"".  5.   361  =  102-^. 

2.  75  =  101-8^  4.    50  =  10i-«9.  6.   45  =  10^-^. 

Write  the  following  as  powers  of  10: 

7.  log  20  =  1.3.  9.  log  3  =  0.47.         11.   log  111  =  2.04. 

8.  log  500  =  2.70.      10.   log  7  =  0.84.         12.   log  21  =  1.32. 

530.  Characteristic  and  Mantissa.  The  integral  part  of  a 
logarithm  is  called  its  characteristic,  and  the  decimal  part  its 
mantissa. 

Tlie  characteristic  of  the  logarithms  of  a  number  greater  than 
unity  is  one  less  than  the  number  of  digits  at  the  left  of  the  deci- 
mal point. 

ORAL   EXERCISES 

1-12.  State  the  characteristic  and  the  mantissa  in  each  of 
the  logarithms  in  Exercises  1-12  above. 

531.  Since  the  characteristics  of  logarithms  can  be  deter- 
mined by  inspection,  tables  of  logarithms  furnish  only  the 
mantissas. 


LOGARITHMS  385 

EXPLANATION   OF  THE  TABLES 

532.  The  use  of  the  tables,  pp.  389  and  390,  is  best  seen 
from  an  example. 

Find  the  logarithm  of  365. 

The  first  column  in  the  table,  p.  389,  contains  the  first  two  figures  of 
the  numbers  whose  mantissas  are  given  in  the  table,  and  the  top  row  con- 
tains the  third  figure. 

Hence,  find  36  in  the  left-hand  column,  p.  389,  and  5  at  the  top. 

In  the  column  under  5  and  opposite  to  36  we  find  5623,  the  required 
mantissa. 

Since  365  is  greater  than  100  (or  lO^)  but  less  than  1000  (or  10^),  the 
characteristic  of  the  logarithm  is  2. 

Therefore,  log  365  is  2.5623. 


WRITTEN     EXERCISES 

By  use  of  the  table  find  the  logarithms  of : 

1.  25.                 5.    99.               9.    9.9.  13.  1000. 

2.  36.                6.   86.             10.   8.6.  14.  5000. 

3.  50.                7.   999.           11.   33,000.  15.  505. 

4.  75.                 8.   800.           12.   99,900.  16.  5.05. 

533.  Negative  Characteristics.  An  example  will  serve  to 
show  how  negative  characteristics  arise : 

From  log  346  =  2.5391,  we  find, 

log  34.6  =  log  —  =  log  346  -  log  10  =  2.5391  -  1  =  1.5391. 

log  3.46  =  log  ^  =  1.5391  -  1  =  0.5391. 

log  .346  =  log  ^^  =  0.§391  -  1. 

In  the  last  line  we  have  a  positive  decimal  less  1,  and  the  result  is  a 
negative  decimal ;  viz.  —  .4609.  But  to  avoid  this  change  of  mantissa,  it 
is  customary  not  to  carry  out  the  subtraction,  but  simply  to  indicate  it. 
It  might  be  written  —  1  +  .5391,  but  it  is  customarily  abridged  into 
1.5391.  The  mantissa  is  kept  positive  in  all  logarithms.  The  logarithm 
1.5391  says  that  the  corresponding  number  is  greater  than  lO-i  (or  ^) 
but  less  than  lO*'  or  1. 


386  A  HIGH  SCHOOL  ALGEBRA 

We  now  write  log    .346  =  1.5391. 

Similarly,         log  .0346  =  log  '^^  =  1.5391  -  1  =  2-5391. 

Thus  we  see  that  the  mantissa  remains  the  same,  no  matter  how  the 
position  of  the  decimal  point  is  changed.  The  mantissa  is  determined 
solely  by  the  sequence  of  digits  constituting  the  number. 

The  characteristic  is  determined  solely  by  the  position  of  the  decimal 
point.  The  characteristics  of  the  logarithm  of  a  number  smaller  than 
unity  is  negative  and  equals  the  number  of  the  place  occupied  by  the  first 
significant  figure  of  the  decimal. 


EXAMPLES 

1.  What  is  the  characteristic  of  log  .243  ? 

.243  is  more  than  .1  or  lO-i  but  less  than  1  or  10°.     Hence, 

.243  =  10"^+^  decimal  ^ 

The  characteristic  is  1. 

2.  Similarly,  since  .00093  is  greater  than  .0001  or  10"*  but 
less  than  .001  or  lO-^, 

log  .00093  =  4  +  a  decimal. 

534.  The  characteristic  having  been  determined,  the  mantissa 
is  found  from  the  table  in  the  usual  way. 

For  example  : 

log     .243  =  1.3866, 
log  .00093  =  4.9685. 

WRITTEN     EXERCISES 

Find  the  logarithms  of: 

1.  .35.  3.    .105.  5.    .0023.  7.   .00342. 

2.  .634.  4.    .027.  6.    .0123.  8.    .0004. 

535.  In  finding  the  number  corresponding  to  a  logarithm 
with  negative  characteristic,  the  same  method  is  followed  as 
when  the  characteristic  is  positive.  The  mantissa  determines 
the  sequence  of  digits  constituting  the  number ;  the  character- 
istic fixes  the  position  of  the  decimal  point. 


LOGARITHMS  387 

]  * 

For  example : 

If  log  n  =  2.5955,  the  digits  of  n  are  394.  The  characteristic  2  says 
that  n  is  greater  than  10-2  (^or  .01)  but  less  than  lO-i  (or  .1).  Hence, 
the  decimal  point  must  be  so  placed  that  n  has  no  tenths  but  some 
hundredths.    Therefore  n  =  .0394. 

WRITTEN    EXERCISES 
By  use  of  the  tables  find  the  numbers  whose  logarithms  are : 


1.   1.6232. 

Y. 

2.0792. 

13.   3.0969. 

2.   1.4914. 

8. 

2.7076. 

14.   1.6972. 

3.    2.4281. 

9. 

4.9196. 

15.   3.9284. 

4.    1.9196. 

10. 

3.2201. 

16.   2.9284. 

6.    0.9196. 

11. 

0.2201. 

17.    1.9284. 

6.    3.4281. 

12. 

1.2201. 

18.   5.7832. 

Find  n  if : 

19.    log  n  = 

1.9289 

21. 

log 

(n-l)  =  3.9294. 

20.   log  n  = 

0.9289. 

22. 

log  (in)  =1.6128. 

USE    OF    THE    TABLES    FOR    COMPUTATION 

536.  For  use  in  computation  by  logarithms  the  laws  of 
exponents  may  be  expressed  thus : 

The  logarithm  of  a  product  is  the 
I.       10"*  •  10'"  =  10"'"'"'*.       sum  of  the  logarithms  of  the  factors. 

The  logarithm  of  a  quotient  is  the 
TT  l^'"_in"»-'-       logarithm  of  the  dividend  minus  the 

\(y  ~  '      logarithm  of  the  divisor. 

TJie  logarithm  of  a  number  with  an 
III.        (lO"')'"  =  lO"***.        exponent  is  the  product  of  the  expo- 
nent and  the  logarithm  of  the  number. 

Since  r  may  be  positive  or  negative,  integral  or  fractional,  Law  III 
provides  not  only  for  raising  to  integral  powers,  but  also  for  finding  recip- 
rocals of  such  powers,  and  for  extracting  roots. 


388  A   HIGH   SCHOOL   ALGEBRA 

EXAMPLES 

1.  Multiply  21  by  37. 

1.  log  21  =  L3222,  table,  p.  390. 

2.  log  37  =  1.5682,  table,  p.  390. 

3.  Adding,  log  21  +  log  37,  or  log  (21  x  37)  =  2.8904  (Sec.  636). 

4.  .'.21  X  37  =  111  from  table,  p.  391. 

2.  Divide  814  by  37. 

1.  log  814  =  2.9106,  table,  p.  391. 

2.  log  37  =  1.5682,  table,  p.  390. 

3.  .-.  log  814  -  log  37  =  2.9106  -  1.5682  =  1.3424. 

4.  .-.  814  -f-  37  =  22,  table,  p.  390. 

3.  Extract  the  cube  root  of  729. 

1.  \/729=(729)i 

2.  log  729^  =  \  log  729  (Sec.  536). 

3.  log  729  =  2.8627,  table,  p.  391. 

4.  ^  of  2.8627  =  0.9542. 

5.  .-.  (729)  ^  or  v^729  =  9,  table,  p.  391. 

WRITTEN    EXERCISES 

Compute  by  use  of  logarithms : 

1.  8  X  15.                    6.    5^  11.  192. 

2.  41  X  23.                  7.    312.  12.  V. 

3.  37  X  17.                  8.    V484.  13.  4*. 

4.  12  X  17.                  9.    v'SiS.  14.  V196. 

5.  893 --19.              10.   940 --47.  15.  ■v'2l6. 

Since  the  logarithm  is  approximate,  the  result  in  general  is  approxi- 
mate. Thus,  log  \/(256)  =  0.60205,  which  is  not  the  logarithm  of  4,  but 
is  sufficiently  near  to  be  recognized  in  the  table. 


LOGARITHMS 


389 


537.   How  to  calculate  the  logarithm  of  a  number  not  found 
in  the  table  is  best  seen  from  an  example. 

EXAMPLE 

Find  the  logarithm  of  257.3  (see  16th  line  of  table,  p.  390)  : 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

25 

3979 

3997 

4014 

4031 

4048 

4065 

4082 

4099 

4116 

4133 

The  numbers  nearest  to  257.3,  whose  logarithms  are  given  in  the  table, 
are  257  and  258.     We  have 

log  258  =  2.4116 

log257^  2.4099 

Difference,     .0017 

That  is,  an  increase  of  1  in  the  number  causes  an  increase  of  .0017  in  the 
logarithm.     Assuming  that  an  increase  of  .3  in  the  number  would  cause 
an  increase  of  .3  of  .0017,  or  .0005,  in  the  logarithm,  then 
log  257 .3  =  2.4099  +  .0005  =  2.4104. 

Notes  :  1.  The  product  of  .3  and  .0017  is  .00051,  but  we  take  only  four 
places,  because  the  mantissas  as  given  in  the  table  are  expressed  to  four 
places  only.  If  the  digit  in  the  fifth  decimal  place  of  the  correction  is 
more  than  5,  we  replace  it  by  a  unit  in  the  fourth  place. 

2.  The  difference  between  two  succeeding  mantissas  of  the  table 
(called  the  tabular  difference)  can  be  seen  by  inspection. 

3.  What  is  written  in  finding  the  logarithm  of  257.3  should  be  at  most 
the  following : 

log  257  =  2.4099  tabular  difference   17 

correction  for  .3  = 5  .3 

log  257.3  =  2.4104  5.1 

4.  The  corrections  are  made  on  the  assumption  that  the  change  in  the 
logarithm  is  proportional  to  the  change  in  the  number.  This  is  suffi- 
ciently accurate  when  used  within  the  narrow  limits  here  prescribed. 


WRITTEN   EXERCISES 

Find  the  logarithm  of : 


1.  1235. 

2.  23.5. 

3.  2.36. 

4.  .0237. 


5.  1425. 

6.  1837. 

7.  6720. 

8.  67.25. 


9.  3.142. 

10.  1.414. 

11.  1.732. 

12.  .6226. 


13.  .4071. 

14.  85.51. 

15.  .0125. 

16.  .4267. 


17.  .3002. 

18.  9009. 

19.  12.02. 

20.  5.008. 


390 


A  HIGH  SCHOOL  ALGEBRA 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

0000 

0043 

0086 

0128 

0170 

0212 

0253 

0294 

0334 

0374 

11 

0414 

0453 

0492 

0531 

0569 

0607 

0645 

0682 

0719 

0755 

12 

0792 

0828 

0864 

0899 

0934 

0969 

1004 

1038 

1072 

1106 

13 

1139 

1173 

1206 

1239 

1271 

1303 

1335 

1367 

1399 

1430 

14 

1461 

1492 

1523 

1553 

1584 

1614 

1644 

1673 

1703 

1732 

15 

1761 

1790 

1818 

1847 

1875 

1903 

1931 

1959 

1987 

2014 

16 

2041 

2068 

2095 

2122 

2148 

2175 

2201 

2227 

2253 

2279 

17 

2304 

2330 

2355 

2380 

2405 

2430 

2455 

2480 

2504 

2529 

18 

2553 

2577 

2601 

2625 

2648 

2672 

2695 

2718 

2742 

2765 

19 

2788 

2810 

2833 

2856 

2878 

2900 

2923 

2945 

2967 

2989 

20 

3010 

3032 

3054 

3075 

3096 

3118 

3139 

3160 

3181 

3201 

21 

3222 

3243 

3263 

3284 

3304 

3324 

3345 

3365 

3385 

3404 

22 

3424 

M44 

3464 

M83 

3502 

3522 

3541 

3560 

3579 

3598 

23 

3617 

36;^ 

3655 

3674 

3692 

3711 

3729 

3747 

3766 

3784 

24 

3802 

3820 

3838 

3856 

3874 

3892 

3909 

3927 

3945 

3962 

25 

3979 

3997 

4014 

4031 

4048 

4065 

4082 

4099 

4116 

4133 

26 

4150 

4166 

4183 

4200 

4216 

4232 

4249 

4265 

4281 

4298 

27 

4314 

4330 

4346 

4362 

4378 

4393 

4409 

4425 

4440 

4456 

28 

4472 

4487 

4502 

4518 

4533 

4548 

4564 

4579 

4594 

4609 

29 

4624 

4639 

4(i54 

4669 

4683 

4698 

4713 

4728 

4742 

4757 

30 

4771 

4786 

4800 

4814 

4829 

4843 

4857 

4871 

4886 

4900 

31 

4914 

4928 

4942 

4955 

4969 

4983 

4997 

5011 

5024 

5038 

32 

5051 

5065 

5079 

5092 

5105 

5119 

5132 

5145 

5159 

5172 

33 

5185 

5198 

5211 

5224 

5237 

5250 

5263 

5276 

5289 

5302 

34 

5315 

5328 

5340 

5353 

5366 

5378 

5391 

5403 

5416 

5428 

35 

5441 

5453 

5465 

5478 

5490 

5502 

5514 

5527 

5539 

5551 

36 

5563 

5575 

5587 

5599 

5611 

5623 

5635 

5647 

5658 

5670 

37 

5682 

5694 

5705 

5717 

5729 

5740 

5752 

5763 

5775 

5786 

38 

5798 

5809 

5821 

5832 

5843 

5855 

5H66 

5877 

5888 

5899 

39 

5911 

5922 

5933 

5944 

5955 

5966 

5977 

5988 

5999 

6010 

40 

6021 

6031 

6042 

6053 

6064 

6075 

6085 

6096 

6107 

6117 

41 

6128 

6138 

6149 

6160 

6170 

6180 

6191 

6201 

6212 

6222 

42 

6232 

6243 

6253 

6263 

6274 

6284 

6294 

6304 

6314 

6325 

43 

6335 

6345 

6355 

6365 

6375 

6385 

6395 

6405 

6415 

6425 

44 

6435 

6444 

6454 

6464 

6474 

6484 

6493 

6503 

6513 

6522 

45 

6532 

6542 

6551 

6561 

6571 

6580 

6590 

6599 

6609 

6618 

46 

6628 

6637 

6646 

6656 

6665 

6675 

6684 

6693 

6702 

6712 

47 

6721 

6730 

6739 

6749 

6758 

6767 

6776 

6785 

6794 

6803 

48 

6812 

6821 

6830 

6839 

6848 

6857 

6866 

6875 

6884 

6893 

49 

6902 

6911 

6920 

6928 

6937 

6946 

6955 

6964 

6972 

6981 

50 

6990 

6998 

7007 

7016 

7024 

7033 

7042 

7050 

7059 

7067 

51 

7076 

7084 

7093 

7101 

7110 

7118 

7126 

7135 

7143 

7152 

52 

7160 

7168 

7177 

7185 

7193 

7202 

7210 

7218 

7226 

7235 

53 

7243 

7251 

7259 

7267 

7275 

7284 

7292 

7300 

7:308 

7316 

54 
N 

7324 

7332 

7340 

7348 

7356 

7364 

7372 

7380 

7388 

7396 

0 

1 

2 

3 

4 

6 

7 

8 

9 

LOGARITHMS 


391 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

55 

7404 

7412 

7419 

7427 

7435 

7443 

7451 

7459 

7466 

7474 

56 

7482 

7490 

7497 

7505 

7513 

7520 

7528 

7536 

7543 

7551 

57 

7559 

7566 

7574 

7582 

7589 

7597 

7604 

7612 

7619 

7627 

58 

7634 

7642 

7649 

7657 

7664 

7672 

7679 

7686 

7694 

7701 

59 

7709 

7716 

7723 

7731 

7738 

7745 

7752 

7760 

7767 

7774 

60 

7782 

7789 

7796 

7803 

7810 

7818 

7825 

7832 

7839 

7846 

61 

7853 

7860 

7868 

7875 

7882 

7889 

7896 

7903 

7910 

7917 

62 

7924 

7931 

7938 

7945 

7952 

7959 

7966 

7973 

7980 

7987 

63 

7993 

8000 

8007 

8014 

8021 

8028 

8035 

8041 

8048 

8055 

64 

8062 

8069 

8075 

8082 

8089 

8096 

8102 

8109 

8116 

8122 

65 

8129 

8136 

8142 

8149 

8156 

8162 

8169 

8176 

8182 

8189 

66 

8195 

8202 

8209 

8215 

8222 

8228 

8235 

8241 

8248 

8254 

67 

8261 

8267 

8274 

8280 

8287 

8293 

8299 

8306 

8312 

8319 

68 

8325 

8331 

8338 

8344 

8351 

8357 

8363 

8370 

8376 

8382 

69 

8388 

8395 

8401 

8407 

8414 

8420 

8426 

8432 

8439 

8445 

70 

8451 

8457 

8463 

8470 

8476 

8482 

8488 

8494 

8500 

8506 

71 

8513 

8519 

8525 

8531 

8537 

8543 

8549 

8555 

8561 

8567 

72 

8573 

8579 

8585 

8591 

8597 

8603 

8609 

8615 

8621 

8627 

73 

8633 

8639 

8645 

8651 

8657 

8(563 

8669 

8675 

8681 

8686 

74 

8692 

8698 

8704 

8710 

8716 

8722 

8727 

8733 

8739 

8745 

75 

8751 

8756 

8762 

8768 

8774 

8779 

8785 

8791 

8797 

8802 

76 

8808 

8814 

8820 

8825 

8831 

8837 

8842 

8848 

8854 

8859 

77 

8865 

8871 

8876 

8882 

8887 

8893 

8899 

8904 

8910 

8915 

78 

8921 

8927 

8932 

8938 

8943 

8949 

8954 

8960 

8965 

8971 

79 

8976 

8982 

8987 

8993 

8998 

9004 

9009 

9015 

9020 

9025 

80 

9031 

9036 

9042 

9047 

9053 

9058 

9063 

9069 

9074 

9079 

81 

9085 

9090 

9096 

9101 

9106 

9112 

9117 

9122 

9128 

9133 

82 

9138 

9143 

9149 

9154 

9159 

9165 

9170 

9175 

9180 

9186 

83 

9191 

9196 

9201 

9206 

9212 

9217 

9222 

9227 

9232 

9238 

84 

9243 

9248 

9253 

9258 

9263 

9269 

9274 

9279 

9284 

9289 

85 

9294 

9299 

9304 

9309 

9315 

9320 

9325 

9330 

9335 

9340 

86 

9345 

9350 

9355 

9360 

9365 

9370 

9375 

9380 

9385 

9390 

87 

9395 

9400 

9405 

9410 

9415 

9420 

9425 

9430 

9435 

9440 

88 

9445 

94^50 

9455 

9460 

9465 

9469 

9474 

9479 

9484 

9489 

89 

9494 

9499 

9504 

9509 

9513 

9518 

9523 

9528 

9533 

9538 

90 

9542 

9547 

9552 

9557 

9562 

9566 

9571 

9576 

9581 

9586 

91 

95^)0 

9595 

9600 

9605 

9609 

9614 

9619 

9624 

9628 

9633 

92 

mss 

9643 

9647 

9652 

9657 

9661 

9666 

9671 

9675 

9680 

93 

9685 

9689 

9694 

9699 

9703 

9708 

9713 

9717 

9722 

9727 

94 

9731 

9736 

9741 

9745 

9750 

9754 

9759 

9763 

9768 

9773 

95 

9777 

9782 

9786 

9791 

9795 

9800 

9805 

9809 

9814 

9818 

96 

9823 

9827 

9832 

9836 

9841 

9845 

9850 

9854 

9859 

9863 

97 

9868 

9872 

9877 

9881 

9886 

9890 

9894 

9899 

9903 

9908 

98 

9912 

9917 

9921 

9926 

9930 

i)934 

9939 

9943 

9948 

9952 

99 

9956 

9961 

9965 

9969 

9974 

9978 

9983 

9987 

9991 

9996 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

26 


392 


A   HIGH   SCHOOL   ALGEBRA 


538.  To  calculate  the  number  whose  logarithm  is  given 
apply  the  table  as  follows : 

(1)  If  the  given  logarithm  is  in  the  table,  the  number  can 
be  seen  at  once. 

(2)  If  the  given  logarithm  is  not  in  the  table,  the  number 
corresponding  to  the  nearest  logarithm  of  the  table  may  be 
taken. 

A  somewhat  closer  approximation  may  be  found  by  using 
the  method  of  the  following  example  : 

EXAMPLE 

Find  the  number  whose  logarithm  is  1,4271.  The  mantissas 
nearest  to  this  are  found  in  the  17th  line  of  table,  p.  389. 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

26 

4150 

4166 

4183 

4200 

4216 

4232 

4249 

4265 

4281 

4298 

The  process  is  the  reverse  of  that  of  finding  the  logarithm. 

The  next  smaller  mantissa  in  the  table  is  4265,  corresponding  to  the 
number  267.  The  difference  between  this  mantissa  and  the  given  man- 
tissa is  6.  The  tabular  difference  between  4265  and  the  next  larger  man- 
tissa is  16.  An  increase  of  16  in  the  logarithm  corresponds  to  an  increase 
of  .1  in  the  number.  Hence,  an  increase  of  6  in  the  logarithm  corresponds 
to  an  increase  of  j%  of  1,  or  .4,  in  the  number.  This  means  .4  of  one  unit 
in  the  number  267.  What  its  place  value  is  in  the  final  result  depends 
upon  the  characteristic.     The  digits  of  the  result  are  2674. 

The  characteristic  1  shows  that  the  desired  number  is  greater  than  the 
first  power  of  10,  but  less  than  the  second  power  of  10  or  100.  Hence, 
the  decimal  point  must  be  placed  between  6  and  7,  and  the  final  result  is 
26.74. 

Notes  :  1.  For  reasons  similar  to  those  of  Note  1,  p.  391,  the  correc- 
tion should  be  carried  to  one  place  only. 

2.   At  most  the  following  should  be  written  .- 


log  n 
mantissa  for  267 


1.4271 
.4265 


tab.  diff.  16 


diff. 


Therefore, 


6 
n  =  26.74. 


LOGARITHMS  393 


WRITTEN     EXERCISES 


EXAMPLE 

3-  =  729. 

U) 

log  S'^  =  log  729. 

(£) 

X  .  log  3  =  log  729.     Sec.  536,  III. 

(3) 

log  729      2.8627      q 
log  3         .4771 

(4) 

36  =  729. 

Find  the  number  whose  logarithm  is : 

1.  0.7305.       4.   2.9023.  7.   1.1962.  lo.  3.9485. 

2.  0.5029.        5.    3.1467.  8.    2.0342.  n.  4.6987. 

3.  1.4682.        6.   3.6020.  9.    3.3920.  12.  2.6376. 

539.  Exponential  Equations.  When  the  unknown  quantity- 
occurs  in  an  equation  as  an  exponent,  the  equation  is  called  an 
exponential  equation. 


1.  Solve: 

Taking  the  log  of  both 

members, 
Or, 

Dividing  by  log  8,        X 
Test  : 

The  test  is  especially  important  because  the  division  in  step  (4)  is  not 
exact.  The  approximation  is  so  close,  however,  that  the  correct  number 
will  always  be  suggested,  and  the  test  will  finally  verify  it. 

It  should  also  be  noted  that  step  (4)  contains  the  quotient  of  two 
logarithms,  which  is  not  the  same  as  the  logarithm  of  the  quotient  j  hence, 
Sec.  536,  II,   is  not  applicable. 

2.  Solve:  8^-2^'  =  512.  (1) 

(7 -2?/)  log  8  =  log  512.  (^) 

^        log  8         .9031  ^  ^ 

.-.  -2y=-4,OTy  =  2.  (4) 

Test  :  87-2-2  =  83  =  512. 


WRITTEN  EXERCISES 
Solve : 

1.  2-  =  512.  4.    8""  =  4096. 

2.  3^  =243.  5.    10^-2  =  1000. 

3.  22^  =  256.  6.    10*' =  500. 


39i 


A   HIGH   SCHOOL   ALGEBRA 


7.  12^  =  20,736. 

8.  96-^  =  243. 

9.  72- =  343. 


10.  15«--  =  3375. 

11.  a^  =  6^-\ 

12.  (f +1  =  e-'. 


13.   Solve  for  x  and  y,  computing  the  values  to  2  decimal 

places : 

13^  =  2". 


REVIEW 


WRITTEN     EXERCISES 

Find   exactly   or   approximately  by  use   of   logarithms  the 
value  of : 


1.    V2.             4.    ^. 

7.    -^756.          10.    (1.03/. 

2.    V5.            5.    V926. 

8.    -t/812.          11.    (1.04)i«. 

3.    V9.             6.    ^^m. 
13     (164)(798). 

9.    (1.5)^           12.    (1.06)^ 

15.    V(624)(598)(178). 

14. 


(732X774),  /(651)(654)(558). 

(731)(671)  ^  Ai  763 


17.  It  is  known  that  the  volume  of  a  sphere  is  f  ttt^,  r  being 
the  length  of  the  radius.  Using  3.14  as  the  approximate 
value  of  TT,  find  by  logarithms  the  volume  of  a  sphere  of  radius 
7.3  in. 

18.  Find,  as  above,  the  volume  of  a  sphere  whose  radius  is 
36.4  ft. 

Calculate  by  logarithms : 

(132)(1837) 

167 

(2076)(379) 

173 

(3059)(349) 

■    (19)(23)(2443) 


22.    ./(^94)(1842). 
V        307 

23.  ^wwm. 


24.    V.0578. 


LOGARITHMS  395 


25. 


26 


QM^.  27.    (217.6) (.00681). 

(700)i  ___ 

[V278.2(2.578)]^ 


(7688)(7719)  28.     ■;;        '   ^  '    _^^ 

V  m8V2497  •  ^-^^231  .  V76.19 


(248)  (249) 

29.  Given  a  =  0.4916,  c  =  0.7544,  and  b  =  c'^-a\     Find  6. 
Suggestion.  b  =  (c—  a)(c  +  a). 

30.  It  is  known  that  in  steam  engines,  the  piston  head's 
average  velocity  (c)  per  second  is  approximately  given  by  the 
formula :  

where  s  denotes  the  distance  over  which  the  piston  moves 
(expressed  in  the  same  unit  as  c),  and  p  the  number  of  pounds 
pressure  in  the  cylinder. 

(1)  Find  c,  if  s  =  32.5  in.,  p  =  110  lb. 

(2)  Find  ^,  if  c  =  15  ft.,  s  =  2.6  ft. 

31.  Solve  for  x  and  y,  computing  decimal  results  to  2  deci- 
mal places : 

2* -6^=0, 
2x+i  _  12^/  =  0. 

SUMMARY 

The  following  questions  summarize  the  definitions  and  proc- 
esses treated  in  this  chapter : 

1.  What  is  a  logarithm  ?  Sec.  528. 

2.  What  is  the  meaning  of  base  in  logarithms  ?        Sec.  529. 

3.  What  is  the  integral  part  of  a  logarithm  called  ?  What 
is  the  decimal  part  called  ?  Sec.  530. 

4.  State  how  to  find  the  logarithm  of  a  product. 

Sec.  536,  I. 
6.   State  how  to  find  the  logarithm  of  a  quotient. 

Sec.  536,  II. 


396 


A   HIGH   SCHOOL  ALGEBRA 


6.  How  is  the  logarithm  of  a  power  found  ?      Sec.  536,  III. 

7.  State  each  of  the  processes  named  in  statements  4,  5,  and 
6  in  the  form  of  laws  of  exponents.  Sec.  536. 


HISTORICAL   NOTE 

It  is  remarkable  that  the  discovery  of  Logarithms  occurred  so  timely, 
for  this  great  machine  of  computation  was  invented  by  Napier  at  the 
beginning  of  the  seventeenth  century,  just  in  time  to  aid  the  new  work  in 
astronomy  and  navigation  ;  Galileo  had  devised  the  telescope  and  Kepler 
was  ready  to  calculate  the  orbits  of  the  planets.  It  is  also  remarkable 
that  Napier  worked  out  the  principle  of  Logarithms  without  the  use  of 
exponents,  and  this  peculiar  method,  which  is  too  complex  to  be  ex- 
plained here,  produced  tables  quite  different  from  those  now  in  use.  As 
soon  as  Napier's  great  work  on  Logarithms  was  published,  Henry 
Briggs,  a  teacher  in  Gresham  College,  London,  hastened  to  visit  Napier, 
and  suggested  the  advantages  of  the  base  10,  and  thus  laid  the  foundation 
of  our  tables  of  common  Logarithms. 

John  Napier,  also  known  as  the  Baron  of  Merchiston,  was  a  Scotchman, 
born  in  1550,  and  published  his  work  on  Logarithms  in  1614,  only  three 

years  before  his  death.  The 
modesty  and  simplicity  of  the 
great  Scottish  philosopher  is 
shown  by  his  attitude  toward 
Briggs,  for  when  informed  that 
the  latter  had  been  obliged  to 
postpone  his  promised  visit, 
Napier  regretfully  replied, 
"  Ah,  Mr.  Briggs  will  not 
come." 

Napier's  discovery,  whose 
importance  is  not  exaggerated 
by  the  claim  that  "  it  doubled 
the  life  of  the  astronomer  by 
shortening  his  labor,"  followed 
immediately  upon  the  general 
acceptance  of  the  Hindoo  nota- 
tion and  the  introduction  by 
Stevin    of    decimal    fractions. 

Thus,  the  seventeenth  century 
John  Napier  '  ,    ^.  .    ^^ 

saw    the     perfection     of    the 

three  greatest  instruments  of  modern  calculation,  the  Hindoo  notation, 

decimal  fractions,  and  Logarithms. 


CHAPTER   XXVIII 
IMAGINARY   AND   COMPLEX  NUMBERS 

540.  Imaginary  Numbers.  The  numbers  defined  in  what 
precedes  have  all  had  positive  squares.  Consequently,  among 
them  the  equation  ic^  =  —  3,  which  asks,  "  What  is  the  number 
whose  square  is  —  3?  "  has  no  solution. 

A  solution  is  provided  by  defining  a  new  number,  V— 3,  as 
a  number  whose  square  is  —3.  Similarly  we  define  V— a, 
a  being  a  positive  number,  as  a  number  whose  square  is  —  a. 

The  square  roots  of  negative  numbers  are  called  imaginary 
numbers. 

541.  If  a  is  positive,  V—  a  may  be  expressed  Va  V— 1. 


Similarly,  V^^Tio  =  V49(-  1)  =  7\/-  1. 

542.   Real  Numbers.     In  distinction  from  imaginary  num- 
bers, the  numbers  hitherto  studied  are  called  real  numbers. 

WRITTEN    EXERCISES 

Express  as  in  Section  541 : 


1.    V-9. 

4.    V-100. 

7.    V-18. 

10.    V-12. 

2.    V-16. 

5.    -V-(54. 

8.    -V-32. 

11.    V-50. 

3.    V-25. 

6.    V-8. 

9.    -V-7. 

12.    -V~75. 

543.   The  positive  square  root  of  —  1  is  frequently  denoted 

by  the  symbol  i;  that  is,  V—  1=  ^ 

Using  this  we  write  : 

v^^^  =  VE'i;    V  — 49  =  ±7  i;  also  V-  75  a"-^&  =  5  a\/Sb  •  i. 

Note.     Throughout  this  chapter  the  radical  sign  is  taken  as  positive. 

397 


398  A  HIGH  SCHOOL   ALGEBRA 

WRITTEN    EXERCISES 
K-ewrite  the  following,  using  the  symbol  i  as  in  Section  543 : 


1.  2  +  V-4.  5.   25-V--25.  9.  12-V^^. 

2.  3-V^=^.  6.    5-V^^.  10.  2V^=300. 

3.  4+V^^.  7.   3  +  V^^.  11.  4V—  (a  +  h). 

4.  5-V-16.  8.    7+V-12.  12.  Va+V-6V. 


13.    -V-6^c.  16.  p^4-V-(i)  +  9) 


14.    a  +  V-(a2  +  a^).  17.    Va?-V-(a  +  6y 


15.   x-\-y  —^—xy'K  18.    Va/'+2/ +  V— (a7  +  2/)''- 

544.  Complex  Numbers.  A  binomial  one  of  whose  terms  is 
real  and  the  other  imaginary  is  called  a  complex  number. 

The .  general  form  of  a  complex  number  is  a  -f-  hi,  where  a 
and  h  may  be  any  real  numbers. 

Note.  Complex  numbers  are  also  simply  called  imaginary,  any  ex- 
pression which  involves  i  being  called  imaginary.  Single  terms  in  vrhich 
i  is  a  factor  (those  vt^hich  we  have  called  imaginary  above)  are  often 
called  pure  imaginaries,  while  the  others  are  called  complex  imaginaries. 
Thus,  V—  2,  SV—  a,  5  i  are  pure  imaginaries  and  1  —  V—  3,  a  —  V—  h 
are  complex  imaginaries. 

545.  Processes    with    Imaginary    and    Complex    Nimibers. 

After  introducing  the  symbol  i  for  the  imaginary  unit  V— 1, 
the  operations  with  imaginary  and  complex  numbers  are  per- 
formed like  the  operations  with  real  numbers. 

I.   Addition  and  Subtraction. 

EXAMPLE 

Add  V^,    -  V=^,    V^^. 

V^  =  3  i. 
-  \/-25  =  -  5  i. 
V^^  =  V3  •  i. 

.-.  the  sum  is  (3  —  5  +  V3)  i  =  —  (2  —  V-S)  i. 


IMAGINARY   AND   COMPLEX  NUMBERS 


399 


Add: 

1.  2i,  Si,  —L 

2.  Vl6t,  —2%. 
2 


WRITTEN    EXERCISES 


6.   3 +  4  I, 


3  i,  5  +  51 


7.  V-9a^,  _V-8a!2. 

8.  V^(^T^,  -V(6  +  cf. 

9.  2V-32a^  3V^^^^8a^,  6V2i 
6.  6-V^2V'=^,  V^^^.    10.  V3i-1,  V2^  +  2,  1-2V2. 


3.  V-16, 

4.  V^=l[,  V^ 


1. 


II.   Multiplication. 

To  multiply  complex  numbers  we  apply  the  relation  that 
V—  1  •  V— 1  =  —  1,  or  i^  =  —  1,  since  the  square  of  the  square 
root  of  a  number  is  the  number  itself. 


Multiply 


1.    V-16  by 


EXAMPLES 


V-  16  =  4V-  l=4i. 


.-.  the  product  is  12(\/3i)2  =  (12) (-  1)  =  -  12. 
This  may  be  written  (4  i)  (3  i)  =  12  i^  =  -  12. 


2.   3-V^=^by  2-V-5. 

3  -  V3  i 

2-V5  2 


-3\/5i  + vT5i2 
6  -  (2\/3  +  3\/5)i  -  VTS. 


a  +  hi  by  o 

-hi. 

a  +  6i 

a-bi 

a2  +  a6i 

-  abi  - 

bh^ 

&2 


546.   a  +  6t  and  a  —  hi  are  called  conjugate  complex  numbers. 


WRITTEN     EXERCISES 


Multiply : 

1.  5  —  3  i  by  5  +  3  1. 

2.  3+V^^by  2+V^^. 

3.  5  -  2  V^^  by  3  H- 2  V^3. 


4.  5  +  V=^by  5-V^^. 

5.  3  _  V^^  by  3  4- 2 V^2'. 

6.  l-V^^by  2  +  3V^=^. 


400 


A  HIGH  SCHOOL   ALGEBRA 


7.  4  4-iby5  — e.  10.    Vr +  3i  by  V^  — 3i. 

8.  a -\- xi  by  a  —  xi.  11.    V— 25by  V— 9by  V^^. 

9.  a2  +  bH  by  a^  —  bH.  12.    V^~a  by  V^  by  -  ci. 

III.    Division. 

Fractions  (that  is,  indicated  quotients)  may  be  simplified  by 
rationalizing  the  denominator  (Sec.  364,  p.  264). 

For  example  : 

J      V^^^  ^"ir^y/^b^  >/7  .  V5(-  1)  ^  V35 

2     2  -f  V3'3^(2+V33)(3  4.VZr5)  ^6  +  3V^^  +  2\/"^^- \/l5 


3_V-5      (3-V-5)(3  + 


"5) 
=  J_(6  +  3- 


9-(-5) 
'3  +  2V^^-Vl5). 


g     x-\-  yi _        (ce  +  yQ^        _x^  +  2 xyi  — 
'    x  —  yi     (x-  yi)  (x  +  ?/0  x^  +  ?/'-^ 


WRITTEN     EXERCISES 

Write  in  fractional  form  and  rationalize  the  denominators : 


1.  ^JZrQ^^2. 

2.  l-^{a-\-xi). 

3.  V^^-^V^=^. 

4.  Vaic^V— a. 

5.  l--(2-V^^). 

6.  4V^=I^-2V^=^. 

13.    (V^=:^+V-5)h-( 


7.  a-7-(<x  — 5f). 

8.  {a-\-bi)-^{a—bi). 

9.  (3H-60-5-(54-40. 

10.  (V3-  9^)-^-(V2-9^). 

11.  (a;-V^^)-5-(a;  +  V^T). 

12.  (a-^^\Ja^:^ 


2 


5-V-2). 


547.    Powers  of  the  Imaginary  Unit.    Beginning  with  i^=  —  l 
and  multiplying  successively  by  i  we  find : 

- 1.  t6  =  ^4  .  1-2  =  t2  =  _  1. 


1*3  =  1*2 


^. 


f  =i^  .  i  =  —  1  .  i  == 


I. 


*^  =  -l(-l)=4-l. 
i  =  I. 


l4=(+l)2  =  +  l. 

I  =  i. 


IMAGINARY   AND   COMPLEX  NUMBERS  401 

548.    By  means  of  the  values  of  i^,  i^,  i^,  any  power  of  i  can 
be  shown  to  be  either  ±i  or  ±1. 

For  example :  i^^  =  i^^  •  i^  =  (i*)!^  .  i^  =  lis  .  is  =  ^3  =  _  i 


^ 


WRITTEN     EXERCISES 

Simplify  similarly : 

• 

1.    i^                   4.    i^\                  7.    i^i 

10.    I'l^. 

2.    f^                  5.    i^\                  8.    i^. 

11.    i"^. 

3.    i'\                  6.    i^\                  9.    2i««. 

12.     2300^ 

Perform  the  operations  indicated  : 

13.    (1  +  iy.            15.    (1  -  iy  .  i\ 

17 

.    (1  +  0  •^^. 

14.    (1      ^y.             16.    f-1  +  ^7. 

V     ^     y 

18 

/-i-3r 

19.    (l+^)  •(l-^7.                      20. 

(1  +  0^ 

~(i-*y. 

IMAGINARIES   AS   ROOTS   OF  EQUATIONS 

549.    Complex  numbers  often  occur  as  roots  of   quadratic 
equations. 

EXAMPLE 

Solve:  x^-\-x-^l  =  0.  (1) 

Completing  the  square,    X^  -\-  X  +1  =  ^—  1.  (3) 

.:x+J=±V^.  (4) 

.'.x  =  -i±lV-S=-l±^VS.i.  (5) 

Test  :  (_  |  ±  ^  V3  •  iy+(-  i  ±  i  V3  .  i)  +  1  =  0. 

WRITTEN     EXERCISES 

Solve  and  test,  expressing  the  imaginary  roots  in  the  form 
a  -\-bi : 

1.  x^-\-5=0.  5.   «2-6aj  +  10  =  0. 

2.  0^2  _,_  2  a;  4- 2  =  0.  6.   m2  +  4m  +  85  =  0. 

3.  iB2-j-2a;  +  37  =  0.  7.   a^ -\- 10  x  +  4:1  =  0. 

4.  cc2_8a;  +  25  =  0.  8.    x^ -\- SO  x  +  234.  =  0. 


402  A  HIGH  SCHOOL  ALGEBRA 

9.  2/2  -  4  2/  +  53  =  0.  18.  a;2  +  a;  4-  5  =  0. 

10.  ^2-6^  +  90  =  0.  19.  6x2  +  3a;  +  l  =  0. 

11.  p2  4.  20p  + 104  =  0.  20.  10  aj2  _  2  aj  +  3  =  0. 

12.  2  x^-^- 4.x +  3  =  0.  21.  7i2_^_j_;I^^() 

13.  3a^-h2aj4-l  =  0.  22.  12  ar^  +  a;  +  l  =0. 

14.  12  ^2  ^  24  =  0.  23.  8  ^2  _|_  ^  ^  6  ^  q 

15.  6w;2^3o  =  o.  24.  7  a? -^x -\-5  =  0. 

16.  0^-0^  +  1  =  0.  25.  1522+5^-1  =  0. 

17.  4i»2  4-4:a;  +  3  =  0.  26.  8  v2_f_3^  +  6  =  0. 

550.  The  occurrence  of  imaginary  roots  in  solving  equations 
derived  from  problems  often  indicates  the  impossibility  of  the 
given  conditions. 

EXAMPLE 

A  rectangular  room  is  twice  as  long  as  it  is  wide;  if  its 
length  is  increased  by  20  ft.  and  its  width  diminished  by  2  ft., 
its  area  is  doubled.     Find  its  dimensions. 

Solution.     1.   Let  x  =  the  width  of  the  room,  and  2  x  its  length. 

2.  Then  (2  a;  +  20)  (x- 2)  =  2.  2  a:-  a;,orx2-8a:+  20=0. 

3.  Solving  {2),x  =  ^±2i. 

■    The  fact  that  the  results  are  complex  numbers  shows  that  no  actual 
room  can  satisfy  the  conditions  of  the  problem. 

WRITTEN     EXERCISES 
Solve  and  determine  if  the  problems  are  possible : 

1.  In  remodeling  a  house  a  room  16  ft.  square  is  lengthened 
on  one  side  a  certain  number  of  feet  and  decreased  on  the  other 
by  twice  that  number.  If  the  area  of  the  room  as  changed  is 
296  sq.  ft.,  what  are  the  original  dimensions  ? 

2.  A  triangle  has  an  altitude  2  in.  greater  than  its  base. 
if  it  has  an  area  of  32  sq.  ft.,  find  the  length  of  its  base. 

3.  If  a  train  moving  x  mi.  per  hour  travels  90  mi.  in  3  5  —  a; 
hours,  what  is  its  rate  per  hour  ? 


IMAGINARY   AND  COMPLEX  NUMBERS 


403 


GRAPHICAL  REPRESENTATION 

551.  We  have  seen  that  positive  integers  and  fractions  can 
be  represented  by  lines. 

Thus,  the  line  AB  represents  3,  and   n  o 

the  line  BC  represents  3|.  A  B  B  C 

Similarly,  we  have  seen  that  negative  integers  and  fractions, 
which  for  a  long  time  were  considered  to  be  meaningless,  can 
be  represented  by  lines. 

0  0      Thus,  the  line  BA  represents  —  3,  and 

"*     B  b"^      '~^     C  tlie  line    CB  repre- 


A 

sents 


31. 


Irrational  numbers  can  also  be  represented 
by  lines. 

'ihus,  in  the  right-angled  triangle  abc,  the  line  ah 
repi  esents  the  \/2. 

y  Like  the  negative   number 

the     imaginary     number    re- 
-   mained  uninterpreted  several 
-     •       -   centuries.     But   this   number 
„  ^  also  can  be  represented  graph- 

Thus,  if  a  unit  length  on  the 
y-axis  be  chosen  to  represent  V  —  1 
or  i,  the  negative  unit  —  V—  1  or 
—  i  should  evidently  be  laid  off  in 
the  opposite  direction.  3V—  1  or 
3  i  would  then  be  represented  by 
OA  and  —  3  i  by  0J5,  as  in  the  figure,  and  others  similarly. 

The  reason  for  placing  V—  1  or  ^  on  a  line  at  right  angles 
to  the  line  on  which  real  numbers  are  plotted  may  be  seen  in 
the  fact  that  multiplying  1  by  V  —  1  twice  changes  + 1  into 
—  1.  On  the  graph  +  1  can  be  changed  into  —  1  by  turning 
it  through  180°.  If  multiplying  1  by  V—  1  twice  turns  the 
line  1  through  180°,  multiplying  1  by  V—  1  once  should  turn 
-f- 1  through  90°. 


y 


404 


A   HIGH  SCHOOL  ALGEBRA 


For  example  : 

1.  Kepresent  graphically  V—  4 : 

V—  4  =  V'4  i  =  2  i ;  this  is  represented  by  a  line  2  spaces  long  drawn 
upward  on  the  y-axis. 

2.  Eepresent  graphically  — V— 3: 

—  V—  3  =  —  VS  i  =—  1.7  i  (approximately)  ;    this  is  represented  by 
a  line  1.7  spaces  long  drawn  downward  on  the  ?/-axis. 


WRITTEN     EXERCISES 

Kepresent  graphically : 

1.  3  i.  5.    —  5  i. 

2.  —2i.  6.    51 

3.  V^^.  7.    V^=T. 

4.  V^^^ie.  8.   -V^12. 


9.  _5V-4. 

10.  —31 

11.  -h2V^=^. 

12.  5V^^. 


Complex      numbers 

-    may     he     represented 

graphically  by  a  modi- 

.    fication    of     the    plan 

used    in    representing 

imaginary  numbers. 

EXAMPLES 

1.   Represent  graph- 
ically 3  +  i. 

To  do  this  3  is  laid  off 
.    on  the  axis  of  real   num- 
bers  (xx'),  and  i  upward 
on  the  axis  of  imaginaries 
(yy').     As  in  other  graphical  work  this  locates  the  point  Pi  which  is 
taken  to  represent  the  complex  number,  3  +  i. 

The  number  V«-^  +  b'^  is  called  the  modulus  of  the  complex  number 
a  +  bi.  As  appears  from  the  figure  OPi  =  V'3'^  +  1=^,  and  hence  OPi 
represents  the  modulus  of  3  +  i. 


IMAGINARY  AND   COMPLEX  NUMBERS  405 

2.  Represent  graphically  3  —  i. 

The  point  P^  is  the  graph  of  the  complex  number  3  —  t,  and  OP2 
represents  its  modulus. 

3.  Eepresent  graphically  —  3  —  5  t. 

The  point  P3  is  the  graph  of  the  complex  number  —  3  —  5  t,  and  OP3 
represents  its  modulus. 

We  have  thus  interpreted  by  means  of  diagrams  positive  and 
negative  integers,  positive  and  negative  fractions,  positive  and 
negative  irrational  numbers,  and  positive  and  negative  com- 
plex numbers;  in  fact,  all  of  the  numbers  used  in  elementary 
algebra. 

SUMMARY 

The  following  questions  summarize  the  definitions  and 
processes  treated  in  this  chapter : 

1.  What  is  an  imaginary  number?  Sec.  540. 

2.  What  term  is  used  to  designate  numbers  not  imaginary  ? 

Sec.  542. 

3.  Define  and  illustrate  a  complex  7iumber.  Sec.  544. 

4.  What  is  the  meaning  of  the  symbol  i  in  complex  numbers? 

Sec.  543. 

6.   How  may  any  integral  power  of  i  be  expressed  ? 

Sees.  547,  548. 
6.   How  do  imaginary  numbers   often   occur   in   practice? 
What  do  imaginary  roots  indicate  in  solving  problems  ? 

Sees.  549,  550. 

HISTORICAL  NOTE 

Negative  numbers  were  so  long  a  stumbling  block  to  mathematicians 
that  their  square  roots  were  naturally  regarded  as  impossible  until  recent 
times.  The  Greeks  understood  irrationals  and  could  express 
many  of  these  numbers  concretely  ;  for  example,  the  \/2  was 
shown  by  them  to  be  the  length  of  the  hypotenuse  of  a  right 
triangle  whose  sides  are  unity,  as  in  the  figure,  but  the  v  —  2 
had  no  meaning  to  them.     The  Hindoo,  Bhaskara,  said : 


406 


A   HIGH  SCHOOL   ALGEBRA 


"The  square  of  a  positive,  as  also  of  a  negative  number,  is  positive,  but 
there  is  no  square  root  of  a  negative  number,  for  it  is  not  a  square." 
Even  the  great  scholars  of  the  sixteenth  and  seventeenth  centuries  did 
little  more  than  to  accept  imaginaries  as  numbers,  and  it  remained  for 
Caspar  Wessel  (1797)  to  make  the  first  concrete  representation  of  complex 
numbers  ;  but  his  discovery  made  little  impression  until  Gauss  emphasized 
its  importance. 

Karl  Friedrich  Gauss  was  born  at  Brunswick,  Germany,  in  1777.     His 
father  was  a  mason  and  took  little  interest  in  his  son's  education,  but 

the  boy's  wonderful  genius  for 
numbers  attracted  the  attention 
of  his  teachers,  who  induced 
the  Duke  of  Brunswick  to  send 
young  Gauss  to  a  preparatory 
school.  He  entered  the  Uni- 
versity of  Gottingen  in  1795  and 
soon  made  discoveries  in  the 
properties  of  numbers  that  won 
for  him  high  rank-among  mathe- 
maticians. On  the  appearance 
of  his  great  work,  Disquisitiones 
Arithmeticoe,  published  in  1801 
when  Gauss  was  only  twenty- 
four  years  old,  his  contemporary, 
Laplace,  declared  Gauss  to  be 
the  greatest  mathematician  of 
all  Europe.  Gauss  died  in  1855 
after  a  life  devoted  to  mathe- 
matics. He  enriched  all  its 
KARL  Friedrich  Gauss  branches,  including  its  applica- 

tions in  astronomy  and  physics,  with  the  lasting  products  of  his  wonderful 
genius.  He  has  been  worthily  called  :  Princeps  Mathematicorum,  the 
Prince  of  Mathematicians. 


CHAPTER   XXIX 


GRAPHS  OF   QUADRATIC   EQUATIONS 

552.  Preparatory. 

1.  By  counting  the 
spaces  read  the  length 
of  EF  in  the  figure. 

2.  Is  it  the  square  of 
the  length  of  OE? 

3.  Answer  similar 
questions  for  OH  and 
OH. 

Every  point  of  the 
curve  is  solocated  that 
the  length  of  its  ordinate 
is  the  square  of  its  ab- 
scissa. 

553.  Quadratic  ex- 
pressions may  be  rep- 
resented graphically. 

For  example : 

The  curve  in  the  figure 
"s  the  graph  of  y  =  x^. 
That  is,  the  length  of  CI) 
is  the  square  of  that  of  — 
OC;  the  length  of  AB  is 
the  square  of  that  of  OA  ;  etc. 


I         j         ;         I         ! 

/ 

1         ; 

..__|.„L_j L-.Lie 

1       : 

___1g__[ 

i      1   T     1      i 

1         i 

::t|H:i 

"~T"T"" 

4f: 

1         !_       V         i         !      9 

!        I 

F      [       J 

^Tp 

4 

*          1            i 

]  .  l-_-J.„Ai. .L_4 

4 

1            i 

\  i 

j            1 

•-4-i--i----r\[--; 

-~-i ; 

/i    i    :    1    V 

J 

c 

t^ 

E    in    i 

^:      1      ;      !      1    0 

^' 

1 

2 

3       14       i^ 

..4._.|-. 
I    j 

ORAL    EXERCISES 

1.  In  !/  =  ar^  what  is  y  when  a;  =  2  ?     Locate  in  the  figure  the 
point  having  these  values  of  x  and  y. 

2.  Answer  the  same  question  when  a;  =  1 ;  also  3;  0;  4. 

27  407 


NUMBEE 

Square 

-5 

25 

-4 

16 

-3 

9 

-2 

4 

-1 

1 

-0 

0 

1 

1 

2 

4 

3 

9 

4 

16 

5 

25 

408  A  HIGH   SCHOOL  ALGEBRA 


WRITTEN   EXERCISES 

1.  Construct  on  a  large  sheet  of  squared  paper  the  points 
corresponding  to  the  table  of  squares  given  below. 

2.  Then  sketch  a  smooth  curve  through  the 
points  beginning  with  —  5,  25. 

The  work  should  be  carefully  done,  and  the  result  pre- 
served for  later  use.  As  there  are  no  negative  values 
of  x^^  the  a:-axis  should  be  taken  near  the  lower  edge  of 
the  paper.  The  unit  chosen  should  be  quite  large ;  for 
example,  10  spaces.  Then  the  table  might  include  squares 
of  numbers  increasing  by  tenths:  1,  1.1,  1.3,  etc.  The 
curve  will  be  a  graphical  table  of  squares  and  square 
roots. 

3.  Eead  to  one  decimal  place  from  the  graph 

V2;  V3;  V5;  V6;  V7;  V8. 

Every  y-distance  is  the  square  of  the  corresponding  cc-distance  ;  and 
every  x-distance  is  the  square  root  of  the  corresponding  ^/-distance.  We 
see  that  for  every  y-distance  there  are  two  corresponding  a;-distances, 
one  plus  and  the  other  minus,  corresponding  to  the  two  square  roots. 
Thus  the  points  of  the  curve  for  which  y  =  4  are  those  whose  values 
of  X  are  2  and  —  2,  respectively,  i.e.  Vi  =  ±  2. 

554.  Graphical  Solution  of  Quadratic  Equations.  Any  value 
of  X  which  satisfies  the  system 

[y  —  —px  —  q 

makes  x^  equal  to  —px  —  qjOvx^-\-px  +  q=  0. 

The  values  of  x  satisfying  the  system  may  be  read  from  the 
graph  ot  y=:  x\ 

EXAMPLE 
Solve  graphically  x^  —  x—6  =  0. 

1.  Construct  the  graph  of  y  =  x^.     (As  in  Sec.  553.) 

2.  Construct  the  graph  of  y  =  x  +  6. 

3.  They  intersect  at  points  for  which  x  =  —  2  and  +  3. 
.-.  the  roots  of  a;2  -  x  -  6  =  0  are  —  2,  3. 


GRAPHS  OF  QUADRATIC  EQUATIONS 


409 


Notes:    1.   Step  2  may  be  done  by  simply  noting  two  points  of  the 
graph  of  y  =  x  +  6  and  laying  a  ruler  connecting  them.     The  roots  can 


^  ••  X 


be  read  while  the  ruler  is  in  position,  and  thus  the  same  graph  for  y  =  a;'-* 
can  be  used  for  several  solutions. 

2.   The  equation  must  first  be  put  in  the  form  a:^  +  j3x  +  g  =  0,  if  not 
so  given. 

WRITTEN    EXERCISES 
Solve  graphically : 

1.  x^-5x-\-6  =  0. 

2.  a;2  +  3aj  +  2  =  0. 

3.  a;2-2a;-3  =  0. 

4.  a-2-f  2i»-3  =  0. 

5.  x'^ -3 X-  4.0  =  0. 

6.  0:24.4  a;  4- 4  =  0. 

7.  2a;2-a;-l  =  0. 

8.  3a;2_2a;-l  =  0. 

9.  a;2  +  a;+^  =  0. 


10.  x^  +  x  —  2  =  0. 

11.  4a;2  +  4a?  +  l  =  0. 

12.  aj2_9=,o. 

13.  072  — 4  a; +  4  =  0. 

14.  ic2  +  |x  +  |  =  0. 

15.  x^-4x  +  3  =  0. 

16.  a^-4:X-5  =  0. 

17.  a^- 7  a; +  12  =  0. 

18.  a;2_3a._io  =  0. 


410 


A  HIGH  SCHOOL  ALGEBRA 


19.    The  path  of  a  projectile  fired  horizontally  with  a  given 
velocity    from    an    elevation,    as    at    0    in   the   figure,    may 

be  represented  by  the  graph  of  the  equation  2/  =  |^,  where 

g  =  S2  and  v  is  the  initial  velocity  of  the  projectile  in  feet  per 
second.  Let  v  =  16  ft.  per  second  and  compute  the  numbers  to 
complete  the  table  of  values  of  x  and  y. 


Tablb 

X    y 


0 

() 
( ) 
( ) 
() 
() 
( ) 
() 
( ) 






\            \            ]            \ 

4 

..•0       h       -2       ^3        4       5        6        7        8        9        10 

W 

_J 

^^^^^^>s. 

1 

2 

3 

>^ 

4 

5 

' 

6 

7 

J 

8 

9 

10 

.  i    ■    ■    r 

i 

1 

1 

: 

fill        •     M  ■: 

Read  from  the  graph  of  this  table  the  horizontal  distance 
traveled  by  the  projectile  when  it  is  4  ft.  below  the  starting 
point. 

20.  Construct  similarly  the  path  of  a  projectile  whose  initial 
velocity  is  32  ft.  per  second. 

21.  A  cannon  of  a  fort  on  a  hill  is  300  ft.  above  the  plane 
of  its  base.  The  cannon  can  be  charged  so  as  to  give  the  pro- 
jectile an  initial  velocity  of  100  ft.  per  second.  How  far  does 
a  projectile  travel  horizontally  before  it  strikes  the  plane  ? 

22.  The  enemy  is  observed  at  a  point  2|-  mi.  from  the  foot 
of  the  vertical  line  in  which  the  cannon  stands.  With  what 
initial  velocity  must  the  ball  start  to  strike  the  enemy  ? 


GRAPHS  OF   QUADRATIC   EQUATIONS 


411 


555.  Graphs  of  Quadratic  Functions.  The  solutions  of 
ax^ -{- bx -{- c  =  0  may  also  be  represented  graphically  by  plot- 
ting the  curve  corresponding  to  y  =  ax^  -\-hx-\-c.  The  real 
roots  of  the  equation  will  be  represented  by  the  points  where 
the  curve  cuts  the  a^axis. 

EXAMPLE 


Represent  the  solutions  of  a;^  -f-  a;  —  2  =  0. 

To  draw  the  curve  y  =  ic^  _j. 
a:  —  2,  we  make  a  table  of  a 
sufficient  number  of  pairs  of 
values  of  x  and  y,  plot  the  cor- 
responding points,  and  sketch  a 
smooth  curve  through  them. 


X 

y 

-4 

10 

-3 

4 

-2 

0 

-  1 

-    2 

3 

-    2t% 

-\ 

-    2i 

-\ 

-    2A 

0 

-    2 

1 

0 

2 

4 

3 

10 

4 

18 

Note.  When  the  integral 
values  do  not  satisfactorily  out- 
line the  shape  of  the  curve,  frac- 
tional values  of  x  must  also  be 
used  as  above. 


=   '•      y 

\  ^   _„    ^„. 

\   \  T 

„^ ,___;. .18 

„.... 

17 



J 

15 

..-_ 

-  -  '/•  •  ■- 

14 

/              ^ 

13 
12 
11 

z. 



_.  /      - 

♦                          10 
1                            ^ 

\                        ^ 
V                   4 

"Yd  "^  2 

i". 

~. 

'»'        '_5-4    -3    -2V-I       0 

l/il     :2    13    !4    ;5    :      :" 

\    \    \    \    \    \    \ 

^_. 

,^.-^..^_.^_-,_..^._^ 

\     '■_     \     \     \     \     \    , 

This  graph  not  only  represents  the  solutions  ofcc^  +  jc  —  2  =  0; 
but,  what  is  far  more  important,  it  represents  the  values  and 
variation  of  the  polynomial  x^  -Y  x  —  2  for  all  values  of  x 
(within  the  limits  of  the  figure).  Thus  EF  represents  the 
value  of  the  trinomial  for  x  —  \^  and  CD  that  for  x-=  —  2\, 


412  A   HIGH   SCHOOL   ALGEBRA 

The  figure  also  tells  us  that  x^  -\-  x  —  2  is  positive  when  x  is 
negative  and  numerically  greater  than  —  2  (algebraically  less 
than  —  2).  That  when  x  is  algebraically  less  than  —  2  and  in- 
creasing, the  trinomial  is  positive  and  decreasing,  reaching 
the  value  zero  when  x  =—  2.  As  x  increases,  the  trinomial 
continues  to  decrease,  becoming  negative,  reaching  its  least 
value  for  x  =—  ^.  It  then  begins  to  increase,  through  negative 
values,  reaching  zero  when  x  =  1.  As  x  increases  further,  the 
trinomial  continues  to  increase,  becoming  positive  and  continu- 
ing to  increase  as  long  as  x  increases.  Owing  to  the  fact  that 
a  quadratic  equation  has  two  and  only  two  roots,  the  graph 
of  a  quadratic  function  must  consist  of  a  curve  having  a 
single  bend. 

The  graph  enables  us  to  read  approximately  not  only  the 
roots  of  x"^  +  X  —  2  =  0,  but  also  those  of  any  equation  of  the 
form  x"^  -\-  X  —  2  =  a,  ii  they  are  real. 

For  example : 

1.  The  roots  oix^  +  x  —  2  =  3  are  indicated  by  the  points  where  the 
line  drawn  parallel  to  the  ic-axis  through  the  point  3  on  the  ?/-axis  cuts  the 
graph. 

2.  The  line  parallel  to  the  a;-axis  through  the  point  —  3  does  not  cut 
the  graph  at  all.  This  tells  us  that  there  are  no  real  values  of  x  which 
make  the  trinomial  x^  -\-  x—  2  equal  to  —  3.  The  equation  x^  -\-  x  —  2 
=  —  3  has  imaginary  roots. 

The  graph  shows  (1)  that  the  equation  x"^  +  x  —  2  =  a  has  two  real  and 
distinct  roots  whenever  a  is  positive ;  also  when  a  ranges  from  zero  to  a 
little  beyond  —2;  (2)  that  it  has  imaginary  roots  when  a  is  less  than  a 
certain  value  between  —2  and  —  3  ;  (3)  at  a  certain  point  the  line  paral- 
lel to  the  X-axis  just  touches  the  gi-aph.  This  corresponds  to  the  case  of 
equal  roots  of  the  equation.  The  value  of  a  in  this  case  may  be  read 
from  the  graph  as  —2^.  This  means  that  the  equation x^  -^x  —  2  =  - 2^, 
or  aj2  -f  a;  +  -^  =  0,  has  equal  roots.  This  may  be  verified  by  solving  the 
equation. 

Note.  Since  even  the  best  drawing  is  not  mathematically  accurate, 
results  read  from  a  graph  are  usually  only  approximately  correct.  The 
closeness  of  the  approximation  depends  on  the  degree  of  accuracy  in  the 
drawing. 


GRAPHS  OF   QUADRATIC   EQUATIONS  413 

WRITTEN    EXERCISES 

Treat  each  trinomial  of  Nos.  1  to  9  below  as  follows : 

(1)  Draw  the  graph  of  the  trinomial. 

(2)  From  the  graph  discuss  the  variations  of  the  trinomial 
as  in  Section  555. 

(3)  From  the  graph  read  approximately  the  roots  of  the 
equation  resulting  from  equating  the  trinomial  to  zero. 

(4)  If  the  trinomial  is  equated  to  a,  read  from  the  graph 
the  range  of  values  of  a,  for  which  the  roots  of  the  equation 
are,  (i)  real  and  distinct,  (ii)  real  and  equal,  (iii)  imaginary. 

(5)  Last  of  all,  verify  those  of  the  preceding  results  which 
relate  to  roots  of  the  equations  by  solving  the  equations. 

1.  x^-^5x  —  6.  4.    a;2  +  4a;-5.  7.   x'^ -^  2x -{- S. 

2.  x^-\-3x-{-2.  5.   3^2 -2  a;.  8.    x2-2a;-l. 

3.  2x''-5x-\-2.         6.   x'^-i,  9.    S  +  2x-xl 

10.  Draw  the  graph  of  the  function  2  x^  -\-  5  x  —  3,  and 
state  how  it  varies  as  x  varies  from  a  negative  value,  numer- 
ically large  at  will,  through  zero  to  a  large  positive  value.  For 
what  values  of  x  is  the  function  positive  ?  For  what  values 
negative  ? 

11.  Draw  the  graph  of  the  function  6  x"^  —  x  —  5,  and  state 
how  it  varies  as  x  varies  from  a  negative  value,  numerically 
large  at  will,  through  zero  to  a  large  positive  value.  For  what 
values  of  x  is  the  function  positive  ?  For  what  values  nega- 
tive ? 

12.  What  is  the  graphic  condition  that  ax^  -{- bx  -{-  c  shall 
have  the  same  sign  for  all  values  of  a;  ?  What  must  therefore 
be  the  character  of  the  roots  of  ax-  -{-  bx  -{-  c  =  0,  if  the  tri- 
nomial ax"^  -\-  bx  -{-  c  has  the  same  sign  for  all  values  of  a;  ? 

Determine  m  so  that  each  of  the  following  trinomials  shall 
be  positive  for  all  values  of  x : 

13.  a;2  4- ma? -f- 5.  14.   3a;2— 5  a; .-fm.        15.   mx'^ -\- 6  x  +  S, 


414 


A   HIGH  SCHOOL  ALGEBRA 


GRAPHS  OF  SIMULTANEOUS  QUADRATIC  EQUATIONS 

556.    Preparatory. 

1.   In  the  same  diagram  construct  graphs  to  represent  the 
equations : 

3^2  4-2/2  =  25, 


(x'  +  y'  =  25, 
[x  —  y  =  —l. 

Compare  the  result  with  this  figure. 


J 1.1  u t 

'f 

..  u:uJ.  U..J 

J 

} 

1, 1  i  lit 

-^^Z      '  '- 

A.  lid  J J 

/\    ..  . 

/  \ 

-- 

T 

/    .    ^ 

^^^^n^^ 

_^-^                                I 

• 

I- : 

\ 

1            .            :            ;            =            1 

2.    Solve  the  system  of  equations  in  Exercise  1. 

Compare  the  values  of  x  and  y  with  the  coordinates  of  the 
intersections  of  the  graphs. 

In   how   many  points   does    the   straight   line  intersect    a 
circle  ? 

How  many  solutions  has  the  given  system  of  equations  ? 


GRAPHS   OF   QUADRATIC   EQUATIONS 


415 


3.    Construct  in  one  diagram  the  graphs  of  the  equation : 

p2  +  ?/2  =  25, 
\xy  =  12. 

Compare  the  result  with  this  figure. 


..„„..J 

\ 

i     \ 

' 

' tmiCLL \ l: 

^A 

["'"'"pXf^^ 

u 

j7j  1  1 

♦        \^ 

^  I 

n |'"""'| 1 

Y^-^  I 

1  j7  1  1  i  1 

\ 

"""1  ( 1 r 1 1 

:!i 1 1  1 1 1  ° 

, 



- 

M 1 1 

J 



i\r 

/ . 

..u.«  - 

n tv" 

"j . 

i     1     1     1     ' 

r             -              ^ 

1 . ,. 

L                1 

1'  •• 

1 

r  1 

\   i 

'              >              1 

4.   Solve  the  system  of  equations  in  Exercise  3. 
Compare  the  values  of  x  and  y  with  the  coordinates  of  the 
intersections  of  the  graphs. 

557.   Graphical   Solution   of   Two   Simultaneous   Equations. 

Every  point  of  the  graph  of  one  of  the  equations  has  coordinates, 
X  and  y,  that  satisfy  that  equation ;  the  points  of  intersection 
are  points  of  both  graphs  and  therefore  have  coordinates,  x  and 
y,  that  satisfy  both  equations.  Hence,  to  solve  two  simul- 
taneous equations  graphically,  draw  their  graphs  and  read  the 
coordinates  of  their  intersections.  The  coordinates  of  each 
point  of  intersection  correspond  to  a  solution.  If  the  graphs 
do  not  intersect,  the  system  of  equations  has  no  real  roots,  all 
of  them  being  imaginary  numbers. 


416 


A  HIGH   SCHOOL   ALGEBRA 


WRITTEN     EXERCISES 

Solve  graphically,  and  test  by  cojnputing  x  and  y : 


x-y  =2. 

2.  a^-/  =  25, 
x  +  y  =  l. 

3.  0.-2  +  2/2=10, 
xy  =  ^. 

4.  0^  +  2/2  =  13, 
0-^  =  6. 

5.  0:2  4-2/2  =  13, 
a:  +  22/  =  l. 

6.  ^  —  y'^  =  by 
3x-y=z7. 


7.  0:2  +  2/2  =  25, 

iB2_2/2  =  7. 

8.  a:2_3a;2/  +  22/2  =  0, 
a^-^3y'  =  16. 

9.  4a;2  +  92/2  =  36, 
o.'2  +  2/'  =  25. 

10.  4:a^-\-9y^  =  36, 
2x-Sy  =  5. 

11.  4iB2_92,2^3g^ 

0^2 +  2/' =  16. 

12.  4a:2_9^2^3g^ 

052/  =  18. 


GRAPHS  OF  HIGHER  EQUATIONS 

558.    The   graphs   of  higher   equations   serve  to  show  the 
character  of  the  roots. 

EXAMPLE 

The  diagram  is  the  graph  of 

y,  OTf{x)  =  a^-Sa:^  +  2x-6. 


-J-y 

i    1   1/  M 

x'\    c 

— f— i-— -^— i--— j— " 

1  :  2  i  3  if  4  i  5  i  a; 

1-1 

i      1     /      1      1 

__J-_2 

1-3 

— .]-4 

:j::|}:::j:::t::: 

1-5 

j  /    1    !    i 

— i— 6- 

'"T7 

pjyi     i     T    1 

\h 

/'      1    i    r 

X 

/(^) 

-2 

-30 

~1 

-12 

0 

-    6 

1 

-    6 

2 

-    4 

3 

0 

4 

66 

The  roots  of  the  equation 

x^-Sa^-\-2x-6  =  0 

I  

are      3,  -  V^^,  +  V-2. 


The  curve  shows  that  3  is  the  only  real  root.     The  downward  bend  at  P 


GRAPHS  OF  QUADRATIC   EQUATIONS  417 

indicates  that  the  other  two  roots  are  imaginary,  for  the  curve  does  not  rise 
high  enough  to  cut  the  axis  again,  so  as  to  make  two  more  real  intersec- 
tions.    The  imaginary  roots  are  —  V— 2  and  +  V  — 2. 

Thus  we  have  an  illustration  of  the  fact  that  imaginary  roots  always 
occur  in  pairs.  Consequently,  a  cubic  equation  can  only  have  two  im- 
aginary roots  and  must  have  either  one  or  three  real  roots.  An  equation 
of  the  fourth  degree  may  have  either  one  pair  or  two  pairs  of  imaginary 
roots  ;  hence,  it  may  have  either  none,  two,  or  four  real  roots. 

If  the  curve  just  touches  the  cc-axis  and  then  turns  away  from  it,  the 
equation  has  an  even  number  of  equal  real  roots  corresponding  to  this 
point  of  contact. 


ORAL    EXERCISES 

1.  How  many  roots  has  a  cubic  equation  ?  If  the  roots  of 
the  cubic  equation  are  all  real  and  unequal,  in  how  many  points 
does  the  graph  cut  the  a;-axis  ? 

2.  Can  a  cubic  equation  have  two  and  only  two  real  roots  ? 

3.  If  the  roots  of  a  cubic  equation  are  2,  2  and  —  5,  in  how 
many  places  does  its  graph  cross  the  ovaxis  ?  What  will  indi- 
cate the  equal  roots  ? 

4.  When  might  the  graph  of  a  fourth  degree  equation  not 
cut  the  a;-axis  at  all  ? 

5.  What  is  the  greatest  number  of  intersections  with  the 
or-axis  possible  for  the  graph  of  a  fifth  degree  equation  ? 

559.  Graphs  of  Radical  Equations.  The  graphs  of  radical 
equations  picture  the  changes  which  account  for  extraneous 
roots. 

EXAMPLES 


1.   Given  a;-V2a^-t-17  =  -3.  (i) 

/^N/I^ransposing  and  ^2  -  6  o:  -h  8  =  0.  (2) 

squaring, 
Then,  (a;-2)(a;-4)  =0.  {S) 

Therefore  the  roots  are  2  and  4,  both  of  which  satisfy  (i). 


418 


A  HIGH  SCHOOL  ALGEBRA 


To  find  the  relation  between  the  graphs  of  equation  (1)  and 
equation  (2),  they  are  plotted  on  the  same  axes. 


In  equation  (1)  y,  or/(x)  =x  —  \/2  a:;2  -f  17  +  3. 
In  equation  (2)  y,  or  /(x)  =  x^  —  6  x  +  8. 


VALUES   FOR   PLOTTING 


Equation  (1) 

X  f(x) 


Equation  (2) 


X 

fix) 

-2 

24 

-  1 

15 

0 

8 

1 

3 

2 

0 

3 

-1 

4 

0 

5 

3 

-2  -4 

-  1  -  2.3 

0  -1.1 

1  -0.3 

2  0 

3  0.1 

4  0 
6  -0.4 

In  finding  values  of  y  always  take  the  positive  sign  of  the  radical. 


Although  the  graphs  of  equa- 
tions (1)  and  (2)  are  entirely 
different,  each  cuts  the  a^axis 
in  points  2  and  4.  This  shows 
why  both  roots  of  equation 
(2)  also  satisfy  equation  (1). 
Hence,  no  extraneous  roots  are 
introduced  in  this  case  by  squar- 
ing. 

The  equations  not  being  of 
higher  degree  than  the  second 
can  not  have  other  roots. 


-rfji 

U-l-J-l-]-12)4__i__ 

-- 

""[  _['  ['6 

r|--ff|-tf-|--|-t- 

.... 

r  1  1 5 

\  !     1     1"    [     1   /|     1    "[     1 

;   1    1 4 

\;    :    1    1    i/T   1    1    i 

1    :    ;  3 

Y   1    1   1   /  i   ]  1  1 

_.l.i-.!2 

\  i   "i  "^  /   1    i"  I  1 

--i-i-u 

!\l    !    !/'    i    i"    i.    i 

X'i-2i-l|0 

1  1  2X3  141/51  6  i     T     1      1 

X 

!   i   1-1 

>^\i/i    r^i^4^  ' 

—\-xjy^ 

,.|.j.|j|]Zp!$. 

a 

\    /  f-4 

;     ;     ;     [    1     1     1     1     ; 

\  /\    1 

i    i    :  1 

y\  \  \  \  \\\\\ 

2.    Given 

Squaring, 
Squaring, 


Va?  —  1  —  V^ 
V(x 


V2 


6. 


(1) 


-l)(x-5)  =0. 
aj2  —  6  a;  +  5  =  0. 


Solving  (3),  x=  1,  5.     But  1  is  not  a  root  of  (1). 
(1),  (2),  (3),  and  1  satisfies  (2)  and  (3). 


(3) 
5  satisfies  equations 


GRAPHS   OF   QUADRATIC   EQUATIONS 


419 


To  find  the  relation  between  the  graphs  of  equations  (1),  (2), 
and  (3),  they  are  plotted  on  the  same  axes. 

In  equation  (1)  y,  or  f  {x)  —y/x  —  l—  Vx  —  5  —  V2  a;  —  6. 
In  equation  (2)  y,  ov f(x)  =y/(x  —  l)(x  —  5). 
In  equation  (3)  y,  or/(x)  =x^  —  6x  +  6. 


Equation  (1) 


VALUES   FOE   PLOTTING 
Equation  (2) 


Equation  (3) 


X 

fix) 

-  1 

Imag. 

0 

it 

1 

n 

5 

0 

6 

-1.2 

8 

-2.2 

10 

-2.9 

X 

fix) 

-2 

-4.6 

-1 

-3.5 

0 

-2.2 

1 

0 

2,3,4 

Imag. 

5 

0 

6 

-2.2 

7 

-3.6 

8 

-4.6 

X 

fix) 

-2 

21 

-  1 

12 

0 

6 

1 

0 

2 

-    3 

3 

-   4 

4 

-    3 

5 

0 

6 

.5 

The  graphs  show  that 
equations  (2)  and  (3)  have 
a  root,  1,  which  equation  (1) 
does  not  have.  Hence,  squar- 
ing in  this  case  has  intro-  ..:..;;..i.5l 
duced  the  extraneous  root  1. 

The  diagram  also  shows  ..T."!..!? 
that  the  real  part  of  the 
graph  of  equation  (1)  ends 
at  5 ;  also  that  equation  (2) 
has  two  real  branches,  one 
extending  downward  from 
X  =5  and  the  other  down- 
ward from  X  =  1.  The  im- 
aginary values  of  the  radicals 
account  for  the  interruptions  in  the  graphs  at  points  1  and  5. 

560.  Since  squaring  may,  or  may  not,  introduce  extraneous 
roots,  as  shown  in  the  above  examples,  all  roots  must  be  tested, 
and  only  those  preserved  ivhich  satisfy  the  given  equation. 


420  A   HIGH   SCHOOL  ALGEBRA 

SUMMARY 

The   following   questions    summarize    the    definitions    and 
processes  treated  in  this  chapter: 

1.  Write  the  equation  whose  graph  expresses  the  square 
roots  of  positive  numbers.  Sec.  553. 

2.  What  is  the  general  shape  of  the  graph  of  a  quadratic 
function  ?     In  how  many  places  can  it  cut  the  a^axis  ? 

Sec.  555. 

3.  What  do  the  intersections  of  a  graph  with  the  a;-axis 
represent  ?  Sec.  555. 

4.  If  the  roots  of  a  quadratic  equation  are  imaginary,  how 
will  the  graph  show  it  ?  Sec.  558. 

5.  How  are  the  solutions  of  simultaneous  quadratic  equations 
represented  graphically  ?  Sec.  557. 

6.  How  do  graphs  show  that  the  result  of  squaring  produces, 
in  general,  a  different  equation  ?  Sec.  559. 

7.  What  must  determine  which  are  extraneous  roots  result- 
ing from  transforming  a  given  equation  ?  Sec.  560. 


CHAPTER   XXX 

QUADRATIC   EQUATIONS 

THEORY 

561.  The  general  form  for  a  quadratic  polynomial  with  one 
unknown  quantity  is  ax^  -{-bx  +  c,  where  a,  6,  and  c  denote  any 
algebraic  expressions  not  involving  x,  and  where  a  is  not  zero. 
If  a  is  zero  the  polynomial  is  linear. 

For  example  :  1.    5  x^  —  7  x  -f  8. 

Here  a  =  5,  6  =  —  7,  c  =  8. 

2.    -Jj^x^  +  ^x-     ^"^ 


2w  +  1  2n-  1 

Here  a  — ,  6  =  3,  c  — 


2w+l  2w-l 

WRITTEN     EXERCISES 

Put  the  following  expressions  into  the  form  ax^  -\-hx-\-c: 

1.  3a;H-5a;(a;-2)H-4(a;2-5).  4.    («  +  g) _ g (x^ -  11). 

2.  7(4a;-l)4-(a;-t-3)(a;-2).  5.    {ax-\-h)(cx  +  d). 

3.  a(bx-\-c)(2dx-\-Ze).  6.    {o? -a)  +  {x' -h). 

7.    (x-[-q){x-{-p)-{x-q){2x-p). 


8     (^^V\  -V^-^^ 
\2     3)      3V8        ay 


9.   x^-\-ab  —  ax  —  h{a-\-x-irX^). 

10.  a;(a;-2)(a;-4)-a^(a;-5). 

11.  (2  a; +  1)2- (3  a; +  1)2+ (4  a; +  1)2. 

12.  (aj-  l)(a;-  2)(a;-  3)-(«+  l)(a;+  2)(a;+  3). 

421 


422  A   HIGH   SCHOOL  ALGEBRA 

562.  Similarly,  every  quadratic  equation  can  be  put  into  the 
form  aa?  +  6a;  +  c  =  0  by  transposing  all  terms  to  the  left  mem- 
ber and  then  putting  the  polynomial  which  constitutes  the  left 
member  into  the  form  aoi?  -{■hx-\-c. 

For  example : 

Given  (mx  -f  3  a)^  =  mx'^  —  5(amx  —  2), 
then  m^x^  +  6  amx  +  9  a^  =  mx^  —  5  amx  -\-  10, 
transposing,  (m^  —  m)x^  -{■  11  amx  +  9  a^  —  10  —  0. 
Hence,  a  =  m^  —  m,  6  =  11  am,  c  =  9  a^  _  lo. 

METHODS  OF  SOLUTION 

563.  General  Solution.  By  solving  the  general  quadratic 
equation  aa^  -\-bx-\-c  =  0,  general  formulas  for  the  roots  are 
obtained. 

Solve :  ax^-\-bx-^c  =  0.       (1) 

Dividing  by  a,  which  is  not  0,  X^  +  —  +  -  =  0.  (S) 


a       a 


&2 


Adding  to  complete  the  ,  i^o  r^i        ^ 

square  and  subtracting  the  ~  ^     '  a  y.2      A.  nl      n  ^   ^ 


Writing  the  second  term  as 
the    square 
root, 

Factoring  (5), 


V        2ay        Ua^      a]  ^^ 


the    square    of  its    square     /  6  \^       I  y/h"^  —  4  acX^ 

root,  '^ 


[^-i-:)-rf^r-  <^> 


^      2a 


-4ao\/    _^A+V5izLi«£U0.         (6) 
a        J\        2a  2a        J 


la      \        2a        I 
Denoting  these  roots  by  r-^  and  rg : 


2a      \        2a        I  2a      V       2a        /  ^  ^ 


_5+V62- 

-4ac 

2a 

_5_V6^- 

-4ac 

2  a 

By  substituting  in  these  formulas  the  values,  including  the 
signs,  that  a,  6,  c  have  in  any  particular  equation,  the  roots  of 
that  equation  are  obtained.     This  is  called  solution  by  formula. 


QUADRATIC   EQUATIONS  423 


Solve : 

Here, 

EXAMPLE 

a  =  3,  &=-9,  c=5, 

a) 

(2) 

and 

_(_9)±V(-9)2-4.3.5 
2.3 

(5) 

_  9  ±  VSl  -  60  _  9  ±  V21 
6                       6 

i4) 

WRITTEN    EXERCISES 
Solve  by  formula : 

1.  x^-4:X  +  4:=:0.  9.    x^-x-\-6  =  0. 

2.  a^-5«  +  6  =  0.  10.    2«2-a;  +  2  =  0. 

3.  ic2-3a;  +  2  =  0.  11.   3  a;2-2  x  +  l  =  0. 

4.  a^-x-X  =  0.  12.    7i»2  +  6a;-4  =  0. 

5.  a^  +  3a;  +  l  =  0.  13.    4a;2_  12  a;  +  9  =  0. 

6.  a^  +  2x-l=0.  14.    3a;2  +  5i»-2  =  0. 

7.  ir2-13a;  +  9  =  0.  15.    5x'-4:X-\-6  =  0. 

8.  2iB2-7a;-3  =  0.  16.    7a^  +  5a;-8  =  0. 

564.  Literal  Quadratic  Equations.  When  any  of  the  coeffi- 
cients of  a  quadratic  equation  involve  letters,  the  equation  is 
called  a  literal  quadratic  equation. 

Such  equations  are  solved  in  the  usual  way. 


EXAMPLES 

1.    Solve:                    ic2-f 6mx  +  8  =  0. 

« 

Completing  the  square,    x^  +  6  mx -\- 9  m^  =  9  m^  —  S. 
.'.  (x  +  3m)2  =  9m2-8. 

...  iK  +  3m^  ±V9m2-8. 

.-.  x  =  -Sm±V9m^-8. 

2.    Solve:                         t^  +  gt-\-h  =  0. 

(^) 

Here,                          a  =  1,    b  =  g,   c  =  h. 
Hence,  by  Sec.  563,                       ^  _  —  .^  ±  v  gr    —  4  A^ 
28                    ,                                             ^ 

(5) 

424 


A  HIGH  SCHOOL  ALGEBRA 


3. 

Solve: 

Here,                          a 
Hence,  by  Sec.  563, 

=  g,   b  =2v,   c  =  —  2s. 

{1) 

{2) 
(3) 

2g     2g     ^      ^ 

n           1 

4sr(- 

-2s) 

(4) 

g    g 

(5) 

=  -{-v±y/v'^  +  2gs). 

(6) 

WRITTEN     EXERCISES 

Solve: 

1.  t''-St  +  24:d  =  9d\ 

2.  ay^-{a-b)y-b  =  0. 

3.  w;2 -f  4  awj -f  a2  =  0. 

4.  v^  —  4:  amv  =  (a^  —  m^)^. 

5.  w;2_a2  =  26(a-w). 

6.  t^  +  at=zk. 

7.  ?/2  +  A:w  +  l  =  0. 

8.  v"^  +  mv  =  1. 

9.  ax^-^bx  +  c  =  0. 
10.    a;2  _^  aa;  +  6  =  0. 


11.  mV  +  2ma;  =  -l. 

12.  ie2  +  2pa;-l  =  0. 

13.  4a;2-4aa;  +  16  =  0. 

14.  aV-f  2aaj  +  5  =  0. 

15.  m2a;2  +  4ma;-6  =  0. 

16.  x^  —  4:ax=:9. 

17.  5aa;2  +  36aj4-263  =  0. 

18.  62aj2_2  5a;  =  ac-l. 

19.  .T2-3aa;  +  10a2  =  0. 

20.  2a;2-3a;  =  a(3-4a;), 


565.   Collected  Methods.     We  have  used  three  methods  of 
solving  quadratic  equations : 

1.   Factoring. 


Equation 

a;2_3a;4-2  =  0. 
a^-(a-\-b)x  +  ab  =  0. 


Factors 

(x-2){x-l). 
(x  —  a)(x  —  b). 


Roots 

a;  =  2,  x  =  l. 
x  —  a,  x=  b. 


2.    Completing  the  squai-e. 

Equation  Solution 

a^  +  jc  +2  =  0.  See  Sec.  371. 


Roots 


QUADRATIC   EQUATIONS  425 

ax^  +  bx-{-c=0.  See  Sec.  371.         ^^  -  &  ±  V&^- 4ac^ 

2a 

3.   Formula. 

Equation  Solution  Koots 

Sx^-^2x-7  =  0.         See  Sec.  563.         x=~^^^^^. 


ax'-\-bx  +  c  =  0.  See  Sec.  563.         ^^~b  ±Vb^-4.ac^ 

2a 


WRITTEN     EXERCISES 

Solve  by  factoring : 

1.  x'^-x-6  =  0.  6.  a;2-a;-30  =  0. 

2.  x''-x-2  =  0.  7.  a;2  +  a;-12  =  0. 

3.  x''-{-x-2  =  0.  8.  a;2-3a;  +  2=:.0. 

4.  a;2  +  a;-6  =  0.  9.  a;^ +  11  a; +  30  =  0. 

5.  x^-\-3x-\-2  =  0.  10.  a;2-7a;  +  12  =  0. 

Solve  by  completing  the  square : 

11.  a;2  +  a;  +  l  =  0.  15.  a;^- 5a;  + 10  =  0. 

12.  a;2_^3 0^+1  =  0.  16.  a;^  -  16a;  +  60  =  0. 

17.  a;2  +  |a;  +  i  =  0. 

18.  a;2  +  7.5  a^- 3.5  =  0. 

24.  a;2  +  l  =  0. 

25.  a;2  +  15a;  +  56  =  0. 

26.  aj2  +  8a;  +  33  =  0. 

27.  a;2-10a;  +  34  =  0. 

28.  2a;2  +  3a;-27  =  0. 

Solve  and  test,  using  whichever  of  the  methods  in  Sec.  563 
seems  most  convenient : 

29.  92/2-4  =  0.  31.  5a;2-4a;  +  4  =  0. 

30.  6a;2_l3x  +  6  =  0.  32.  t^ +  llt -\-30  =  0. 


13. 

a;2_|a;  +  l  =  0. 

14. 

«2-1.5x  +  .5  =  0. 

Solve  by  formula : 

19. 

3  a;2  +  a;  +  5  =  0. 

20. 

2x''-5x-S  =  0. 

21. 

4a;2  +  3a;-l  =  0. 

22. 

5a;2^2a;  +  6  =  0. 

23. 

a;2  +  a;  + 1  =  0. 

426  A  HIGH   SCHOOL   ALGEBRA 

33.  6s2-5s-6  =  0.  39.  a;^  -  .7  a; -f  .12  =  0. 

34.  6r2-2r-4  =  0.  -         40.  11  x"" -{- 1  =  4.(2  -  xy. 

35.  ^2  +  4^  — 3  =  0.  41.  x^+{a-\-b)x-{-ab  =  0, 
36  ^a;-l      2a;  +  1^3  42.  a;^ - (6  +  c)a;  +  6c  =  0. 

2a;  +  l      2a;-l        '  43.    2aa;  +  (a  -  2)x- 1  =0. 

37.  a;2_2a;4-3  =  0.  ^2  ^2 

38.  .^2  -  0.5  a;  +  0.06  =  0.  *    b-^x~b-x'^^' 

45.  The  product  of  two  consecutive  positive  integers  is  306. 
Find  the  integers. 

Solution. 

1.  Let  X  be  the  smaller  integer. 

2.  Then  x  +  1  is  the  larger. 

3.  .'.  x(x  +  1)  is  their  product. 

4.  .-.  x(x-^l}  =  306,  by  the  given  conditions. 

5.  .'.  x2  +  X  -  306  =  0,  from  (4). 

6.  ...  X  =  -l^Vr  +  1224  ^  -1±35  ^  ,7^  ^^  _  ,g^  ^^^^.^^  ^,^ 

^  2 

Since  the  integers  are  to  be  positive,  the  value  —  18  is  not  admissible. 
x  =  17,  .'.  x  +  1  =  18,  and  the  integers  are  17  and  18. 
Test.     17  •  18  =  306. 

46.  There  is  also  a  pair  of  consecutive  negative  integers 
whose  product  is  306.     What  are  they  ? 

47.  If  the  square  of  a  certain  number  is  diminished  by  the 
number,  the  result  is  72.     Find  the  number. 

48.  A  certain  number  plus  its  reciprocal  is  —  2.  What  is  the 
number  ? 

49.  A  certain  positive  number  minus  its  reciprocal  is  f. 
What  is  the  number?  What  negative  number  has  the  same 
property? 

50.    The  perimeter  of  the  rectangle 
X     shown  in  the  figure  is  62  in.     Find  the 
sides. 

51.  One  perpendicular  side  of  a  certain  right  triangle  is  31 
units  longer  than  the  other ;  the  square  of  their  sum  exceeds 
the  square  of  the  hypotenuse  by  720.     Find  the  sides. 


QUADRATIC   EQUATIONS  427 

52.    In  a  right  triangle  of  area  60  sq.  ft. ;  the  difference 
between  the  perpendicular  sides  is  7.     Find  the  three  sides. 

Note.    Those  who  have  studied  geometry  may  take  up  some  of  the 
problems  based  upon  geometric  properties  found  in  Chapter  XXXIII, 


RELATIONS   BETWEEN   ROOTS   AND    COEFFICIENTS 

566-   Relation   of    Roots   to   Coefficients.      By   adding   the 
values  found  for  the  roots  (Sec.  563),  we  obtain 

^'"^'*'~     2a  2a  2a  2a  a' 

Multiplying  the  values,  we  find 


_  _5^_V62^-4ac      -b  +  V5^  -  4:  ac  __b^  -  b"^  -\-  4.  ac  __c 
''''''~  2a  2a  ~~         4  a^         ~a 

Applying  these  results  to  the  equation  x^  +  pa;  +  g  =  0,  we 
have : 

^1  +  ^2  =  -i>» 
r,r2  =  q. 

In  words : 

In  the  equation  x^  -f  px  -|-  q  =  0,  the  coefficient  of  x  with  its  sign 
changed  is  the  sum  of  the  roots,  aiid  the  absolute  term  is  their 
product. 

Every  quadratic  equation  can  be  put  into  the  form  x"^  -\- px -\- q  =  (i  hj 
dividing  both  members  by  the  coefficient  of  x'^. 

567.  Symmetric  Functions  of  the  Roots.  It  is  apparent  that 
the  equations  in  Section  bQ>Q  remain  unchanged  if  r^  and  r^  are 
interchanged.  On  this  account  the  expressions  r-^  +  r^  and  r^r^ 
are  called  symmetric  functions  of  the  roots  of  the  quadratic 
equation.  There  are  other  such  functions,  but  these  only  will 
be  treated  here.  The  two  following  sections  show  some  of  their 
uses. 


428  A  HIGH  SCHOOL  ALGEBRA 

568.  By  means  of  Section  566  a  quadratic  equation  may  be 
written  whose  roots  are  any  two  given  numbers. 

EXAMPLES 

1.  Write  an  equation  whose  roots  are  2,  —  3. 

__p  =  ri  +  r2  =  2+(-3)  =  -L     .-.  j^  =  1. 

g  =  rir2  =  2(-3)  =  -6. 
.-.  cc2  -f  a;  —  6  =  0  is  the  equation  sought. 

2.  Write  an  equation  whose  roots  are  -|  +  V—  3,  ^  —  V— 3. 

g  =rir2  =a  +  V:^)a  -  V^=^)=  i  -(-  S)  =  ~'^-. 
.'.  fl;2  —  X  4-  V  =  0  is  the  equation  sought. 

WRITTEN    EXERCISES 

Write  the  equations  whose  roots  are : 

1.   4,  5.  6.   24,30.  11.  a,  -b. 

12.  8,  -40. 

13.  a  —  hi,  a  +  bi. 

14.  l  +  2z,  1-2 1. 

15.  |-V2;i4-V2. 

569.  Testing  Results.  The  ultimate  test  of  the  correctness 
of  a  solution  is  that  of  substitution;  but  this  is  not  always 
convenient,  especially  when  the  roots  are  irrational.  In  such 
cases,  the  relations  between  the  roots  and  coefficients  are  of  use. 

For  example  :  Solving  2x^  —  dx-\-6  =  0, 
or  x2  —  f  X  -f  3  =  0, 

the  roots  are     n  =  |  +  JV—  23  and  r2  =  i  —  JV—  23. 

Adding,  —  (n  +  7*2)  =  —  V-  =  —  f,  the  coefficient  of  x. 

Multiplying,  rir2  =(f)2-(i\/-  23)2=  2 5  ^  ||  _  3^  the  absolute  term. 

Therefore,  the  roots  are  correct.     (Sec.  566.) 

570.  In  what  follows,  the  coefficients  a,  b,  c  are  restricted  to 
rational  numbers. 


2. 

hi- 

7. 

8f ,  10. 

3. 

7, -If 

8. 

-  5,  -  20. 

4. 

-4,  +4. 

9. 

-f  ±iV5. 

5. 

|±V-5. 

10. 

f±iV-47. 

QUADRATIC   EQUATIONS  429 

571.  Character  of  the  Roots.     By  examining  the  formula  for 

the  roots, ^-^ — — — —,  it  appears  that  the  character  of  the 

2  a 

roots  as  real  or  imaginary,  rational  or  irrational,  equal  or  un- 
equal, depends  upon  the  value  of  the  expression  6^  _  4  ac. 

1.  It  b^  —  4  ac  is  positive,  the  roots  are  real. 

Thus,  in  x2  +  4  a;  -  3  =  0,  fe-'^  -  4  ac  =  16  +  12,  or  28,  .-.  the  roots  are 
real  and  unequal. 

2.  If  .62—  4ac  is  a  perfect  square,  the  indicated  square  root  can  be 
extracted,  and  the  roots  are  rational. 

Thus,  in  a:2  _  4  ^  ^  3  —  0,  b^  —  4ac  =  lQ—  12,  or  4,  .♦.  the  roots  are 
rational  and  unequal. 

3.  If  62  _  4  Qjc  is  not  a  perfect  square,  the  indicated  root  cannot  be 
extracted  and  the  roots  are  irrational. 

Thus,  in  x^  +  5  x  +  1  =  0,  52  _  4  ^j^  =  25  —  4  =  21,  .-.the  roots  are 
irrational. 

4.  If  52  _  4  ^c  =  0,  the  radical  is  zero,  and  the  two  roots  are  equal. 
Thus,  in  x2  -  10  x  +  25  =  0,  62  _  4  ojc  =  100  -  4  .  25  =  0,  .  •.  the  roots 

are  equal. 

5.  If  62  —  4  ac  is  negative,  the  roots  are  imaginary. 

Thus,  in  2  x2  —  x  +  1  =  0,  62  —  4  ac  =  1  —  8,  or  —  7,  .♦.  the  roots  are 
complex  numbers. 

Consequently,  it  is  merely  necessary  to  calculate  62  —  4  ac  to  know  in 
advance  the  nature  of  the  roots  of  a  quadratic  equation. 

572.  Discriminant.  Because  its  value  determines  the  char- 
acter of  the  roots,  the  expression  6^  _  4  ac  is  called  the  dis- 
criminant of  the  quadratic  equation. 


ORAL  EXERCISES 

Without  solving  the  equations,  find  the  nature  of  the  roots  of ; 

1.  aj2  +  ic-20  =  0.  8.    6x2  +  aj-l=0. 

2.  x^-\-x-3  =  0.  9.    7a;2  +  3a;-4=0. 

3.  2a;2-a;-t-2  =  0.  10.    -4.X+ Sx'' -\-l  =  0. 

4.  Sx'^-x-\-S  =  0.  11.   5+4a;2-3x==0. 

5.  2i»2  +  2ic-4  =  0.  12.    7x-\-6-^x^  =  0. 

6.  5£c2-3a;  +  6  =  0.  13.    -6x -^9  x'^-\-S  =  0, 

7.  3a;2_4a;-f  5  =  0.  14.   x"^- 6x -\- 4:  =  0. 


a  a 


430  A   HIGH   SCPIOOL   ALGEBRA 

573.  The  relation  x^+px  +  q^x^ —{rT,-\-r2)x-{-i\r2  may  be 
written : 

(1)  x^  -^px  +q  =  {x  —  r^){x  —  r^). 

And  since  x"" -\- px -{- q  =  cta;^  +  ^^  +  c  •  ^  ^j^- ^^         b 

a 
we  have 

(2)  ax"^ -\-bx-{-c  =  a(x  —  r-^)  (x  —  r^. 

574.  The  solution  of  a  quadratic  equation,  therefore,  enables 
us  to  factor  every  polynomial  of  either  form  (1)  or  (2). 

Since  r^  and  rg  involve  radicals : 

1.  The  factors  will  generally  he  irrational. 

2.  The  factors  will  he  rational  when  r^  and  v^  are  so ;  that  is 
when  b^  —  4  ac  is  a  perfect  square. 

3.  TJie  two  factors  involving  x  will  he  the  same  when  the  roots 
are  equal;  that  is,  when  b^  —  4  ac  =  0. 

In  the  last  case  the  expressions  are  squares  and 

(1)  becomes  {x  —  7\y,  and 

(2)  becomes  [  Va  (x  —  rj)]^. 

EXAMPLES 

Trinomial  b^  —  iac  IN" atitre  of  Factors 

1.  3cc2-7x  +  2  49-4.3.2  =  25  rational  of  1st  degree. 

2.  3  ic2  _  7  ic  +  .3  49  -  4  •  3  .  3  =  13  irrational. 

3.  2r;-^-8x  +  8  64-4.2.8  =  0  equal. 

ORAL    EXERCISES 

By  means  of  the  above  test,  select  the  squares ;   also  the 
trinomials  with  rational  factors  of  the  first  degree : 

1.  8a^-8a;  +  2.         5.    o(y'-}-3x-2.  9.    6a^  +  5a;-4. 

2.  ^  +  4?/  +  12.  6.    a^x^ -\- 2  ax -^1.       10.    6x^-5x  +  9. 

3.  30^  +  30^  +  1.         7-    4a.-2  +  4a;  +  l.        11.    40.^-4^-3. 

4.  3z^-\-2z-\-12.        8.    a)2-8i«  +  15.         12.    Sa^-^-Qx  +  S. 


QUADRATIC   EQUATIONS  431 

575.    General  Method   of  Factoring    Quadratic   Trinomials. 

The  actual  factors  of  any  quadratic   trinomial  of   the   form 
ax^  -f-  hx  -f  c  can  be  found  by  solving  the  quadratic  equation : 

ao(?  -{-hx-\-G  =  0, 

and  substituting  the  roots  in  the  relation  : 

ax^  ■\-hx-\-c  =  a(x  —  Vi)  (x  —  rg). 

EXAMPLES 

1.  Factor:  6  a;^ -|_  5  a;  -  4.  (2) 

Solving  6  x2  +  5  a;  -  4  =  0,  X  =  —  f ,  (2) 

Therefore,  6  X^  +  6  X  -  i  =  6  (x  +  i){x  -  ^).  (4) 

Using  the  factor  6  as  3  •  2,  the  result  may  be  written  (3  a;  +  4)  (2  a;  —  1). 

2.  Factor:  a^_a;  +  l.  (1) 

Solving  x^-x  +  1^0,  x  =  i  +  l  V^^.  {2) 

'  Substituting  these  values 
of  X  for  r^  and  Vz, 

Thus,  we  have  a  general  method  of  factoring  which  does  not  depend 
upon  the  trial  methods  of  Chapter  XII. 


WRITTEN    EXERCISES 


Factor 


1.  3.T2_2a^_5.  5.  10w^-12w-\-2.  9.  6x^-7x-\-3. 

2.  9x^-3x-6.  6.  9v2_-L7^_2.  lo.  5a^-40x  +  6. 
S.  6i/  +  y  —  l.  7.  6  0^  +  25  07 +14.  11.  a^"*  —  2  a'"  —  3. 
4.  157/2-47/-35.  8.  222  +  52  +  2.  -12.  c' -13  c' -\- 36. 


432  A   HIGH   SCHOOL   ALGEBRA 

RADICAL    EQUATIONS 

576.  The  simpler  forms  of  radical  equations  can  be  solved 
by  squaring  both  members,  but  other  forms  require  special 
methods. 

EXAMPLE 


Solve:         5x'^-'Sx-\-V5x^-Sx-\-2==lS.        ■  (1) 

Putting  5  x2  -  3  X  =  J/, 

the  given   equation                               , —       ^_  ,^^ 

becomes                                       y  -\- Vy  -^  2  =  18.  {2) 

Subtracting  y,                                           V?/  +  2  =  18  —  y.  {3) 

Squaring,                                                        ^  +  2  =  324  -  36  y  +  ?/2.  {4) 

Kearranging,                            j,2  _  37  ^  +  322  =  0.  (5) 

,.      37  i  V372-4  .  322  ,-^, 

Solving,                                                              y  = — (^) 

=  23  or  14.  (7) 

Hence  the  values  of  x                      e    o       o           f^o  ^  on 

are  determined  from                      b  X-^  —  6  X  =  26,  {8) 

and                                           5  a;2  _  3  x  =  14.      (9) 

Solving  these,                                                    X  =  ^  ^  in  ^^'  ^^^^ 

and                                                                     X  =  2,  or  —  1.4.  (11) 
Test.     Substituting,  it  appears  that  2  and  —  1.4  satisfy  the  equation. 

The  values  ^  ±  ^^^^  satisfy 
10  ^ 

6  a;2  _  3  X  -  \/5  a;2  -  3  a;  +  2  =  18. 

577.    Radical  equations  in  quadratic  form  are  more  easily 
handled  in  the  notation  of  exponents. 


EXAMPLE 

Solve :                              ic"^  +  x-^  -  6  =  0.  (1) 

Eearranglng,                              X"!  +  X~^  -  6  =  0.  (^) 

Factoring,                       (x~^  +  3)  (x"^  -  2)  =  0.  (3) 

Therefore,                        X~^  =—3,  and  X~^  =  2.  (4) 

Or,                                          aj  —  i,  and  X  =  l-  (5) 
Substituting  in  step  (1),  ^  satisfies  the  equation,  but  ^  is  an  extraneous 
root. 

578.    If  the  factors  of  a  quadratic  form  are  not  readily  ap- 
parent, the  quadratic  formula  may  be  applied. 


QUADRATIC   EQUATIONS  433 

579.   If  the  equation  contains  denominators  irrational  in  the 
unknowns,  it  is  generally  best  to  clear  of  fractions. 


Solve : 


WRITTEN    EXERCISES 


1.  6  a;  -  a;^  -  12  =  0. 

2.  x^  -x^  -6  =  0. 

3.  0)2  +  ic  +  Va;2  +  a;  +  1  =  —  1. 

4.  x^-^a-\-  Vaj2  _|.  2  a  =  -  a. 


5.  x^  +  6x  —  5=  Va;2  +  6  x  +  7. 

6.  Va;2  _^  6  aj  -  16  +(a;  +  3)^  =  25. 

7.  2  a;- 5  a;^ +  2  =  0. 

8.  a;~^  -  a;"^  -  6  =  0. 

9.  3  a;"^  —  4  a;"^  =  7. 

10.  a;2  _^  3  a;  -  3  +  Va;^  -f-  3  a;  +  17  =  0. 

11.  a;+V^^^  =  ^-^^^^^. 

x~^x^-S 

12.  ^x(a  +  a;)  +  Va;(a  ~  a;)  =  2  Vaa;. 

13.  V(V1  +  a;2  +  a;)-^(Vl  -\- x^  -  x)  =  4:. 
2a2 


14.  a;+Va^  +  a;2=      , 

Va^  +  a;^ 

15.  V2¥Tii-V^"^^-V^T2  =  o. 

16.  V2/  +  1  4- V2/  -  2  =  V2  2/  +  3. 

/a;  ,   a  ,      Ix      a        14:  x      2  a 

18.    V2  2/  —  2  +  V^  =  V6  2/  —  5. 


19.   2a=V2aa;  +  5a2_V2aa:-3a2. 

Va^2  +  a2  --  Va;2  -f-  6^ 


434  A   HIGH   SCHOOL   ALGEBRA 

CERTAIN  HIGHER  EQUATIONS  SOLVED    BY  THE  AID   OF 
QUADRATIC  EQUATIONS 

580.  We  have  found  the  general  solution  of  linear  and 
quadratic  equations  with  one  unknown.  Equations  of  the 
third  and  the  fourth  degree  can  also  be  solved  generally  by- 
algebra,  and  certain  types  of  equations  of  still  higher  degree 
as  well ;  but  these  solutions  do  not  belong  to  an  elementary 
course.  We  shall  take  up  only  certain  equations  of  higher 
degree  whose  solution  is  readily  reduced  to  that  of  quadratic 
equations. 

EXAMPLES 

1.  Solve:  a;«- 3  0^-4  =  0.  (1) 

Let  y  =  £b8,  then  y^  -  3  y  -  4  =:  0.  {£) 

Solving  (2),  2/  =  4, 

and  y=—l.  (3) 

.-.  by  the  substitution  in  (2),  X^  =  4,  or  x^  —  4  =  0, 

and  a;3  =  —  1,  or  x^  +  I  =  0.  (4) 

Factoring  (A),    x^  -  4  =  (x  -  \/4)  (^2  +  v"!  •  X  +  y/I^)  =  0,  (5) 

and  X3  +  1  =(X  +  1)(X2-X+  1)  =  0.  {6) 

Solving  (5),  a;=  v^i,  or  v^( -  i  +  i  V^^) ,  or  v/4(  -  i—  i  V^^).      (7J 
Solving(6),x=- 1,  or^- IV^  ori  +  ^V^^).  (8) 

2.  Solve:  {x' -3  x +  l)(x' -S  x-^2)=12.  (1) 

This  may  be         (^2  -  3  X  +  1)  [(x2  -  3  X  +  1)  +  1]  =  12.  (^) 

VFritten 

for 
4-  1.  the 

(3) 
(4) 

(6) 
(7) 

These  are  the  four  roots  of  the  given  equation  of  the  fourth  degree. 


If  y  is  put  for 

032  -  3  85  +  1,  the 
equation  becomes 

y(y  +  l)=12, 

or 

y2^y-l2  =  0. 

Solving, 

y  =  Sor  -4. 

Then  ft-om  (S), 

^2  _  3  X  +  1  =  3, 

and 

a;2_3x  +  l=-4. 

Solving  (6), 

2 

Solving  (7), 

.=3±V-U 

QUADRATIC   EQUATIONS  435 

WRITTEN     EXERCISES 

Solve  as  above  : 

1.  a^-3a;2  4-lz=0.  9.   aa^"  -  &a;"  +  c  =  0. 

2.  12  -  a;4  =  11  a;2.  Ill 

10.    ^A-T  + 


3.   x'+ax'-Sa^^O.  a^  +  1      a^-{-2     a^ -\- 3 

4    a.2  ,  5^_A_.  11.   x^^-4:X^-5  =  0. 
x^  +  3 

5.  ^_7^  +  6^0.  12.   2^-^5x^  +  2  =  0. 

6.  a;«- 3  0.-^  +  2  =  0.  13.    (ar^+4)2_4(ar^+4)4-4=0. 

7.  x^^-5x^-}-6  =  0.  ^        ,  1 

14.   a;2^3a;=l- 


8.   a;4  +  13a^+36=0.  a;2  +  3a;+l 

15.  (a^-3a;  +  l)(a^-3a;4-2)  =  12. 

16.  (a;2-f  5a;-l)(a;2^5^^1)^_l^ 

581.  Binomial  Equations.  Equations  of  the  form  x""  ±a  =  0 
are  called  binomial  equations.  The  simpler  cases  admit  of 
being  solved  by  elementary  processes. 


EXAMPLES 

olve: 

a!3-l  =  0,  ora.'«  =  l. 

W 

Factoring  x^  —  I, 

(^-1)(X2  +  X+1)=0. 

(^) 

Finding  equations 
equivalent  to  (2), 

x-1,  x:^  +  x  +  i  =  o. 

(5) 

Solving  (3) 

..1,   ..-i±/-'^. 

U) 

Thus  we  have  found  the  three  numbers  such  that  the  cube  of  each  is  1, 
or  the  three  cube  roots  of  unity. 

Verify  this  statement  by  cubing  each  number  in  step  (4). 

.  1  _  y/ZTs      _  1  _  iy/s 


Note  that     (  -  i  +  i  V-  3) 


2  2 

also  that         (-  i  -  1  V^TS)^  =  '  ^  +  ^^  =  ZllilM. 

Hence,  if  w  denotes  one  of  the  complex  cube  roots  of  1,  w^  is  the  other. 
Every  number  has  three  cube  roots  ;  for  example,  the  cube  roots  of  8 
are  2,  and  2  w,  and  2  a>2. 


436                    A 

HIGH   SCHOOL   ALGP:BRA 

2.    Solve: 

a;^  +  1  =  0,  or  a;^  =  -  1. 

(^) 

Factoring  x*  +  l, 

{x^-i){ai'  +  i)  =  0. 

(^) 

Solving  (2), 

X^=:i,  x^=-l. 

(3) 

Solving  (3), 

X  =  ±  Vi,  X  =  ±  v^—  i. 

(4) 

These  are  the  four  numbers,  each  of  which  raised  to  the  fourth  power 
equals  —  1  or  the  four  fourth  roots  of  —  1. 


WRITTEN    EXERCISES 

1.  Find  the  3  cube  roots  of  —  1  by  solving  oc^  -\-l  =  0. 

2.  Find  the  4  fourth  roots  of  1  by  solving  cc'*  —  1  =  0. 

3.  Find  the  6  sixth  roots  of  1  by  solving 

4.  Find  the  4  fourth  roots  of  16  by  solving  x*  —  16  =  0. 

5.  Find  the  3  cube  roots  of  8  by  solving  cc^  —  8  =  0. 

6.  Show  that  the  square  of   either  irrational  cube  root  of 
—  1  is  the  negative  of  the  other  irrational  cube  root. 

7.  Show  that  the  sum  of  the  three  cube  roots  of  unity  is 
zero ;  also  that  the  sum  of  the  6  sixth  roots  of  unity  is  zero. 


SIMULTANEOUS    QUADRATIC   EQUATIONS   WITH   TWO 
UNKNOWNS 

582.  The  simpler  cases  in  which  one  equation  is  linear  or 
homogeneous  of  the  second  degree,  or  in  which  the  equations 
are  symmetric,  have  been  solved  in  Chapter  XXIII.  The 
following  Sections  583,  584  treat  of  special  cases,  showing 
how  these  may  be  reduced  to  the  simpler  systems. 

A  system  of  tivo  simultaneous  quadratic  equations  whose  terms 
are  of  the  second  degree  in  x  and  y  loith  the  exception  of  the  ab- 
solute terms  can  he  solved  hy  reducing  to  a  system  in  which  one 
equation  is  entirely  homogeneous.     (Sec.  120.) 

This  can  be  done  in  two  ways : 

1.  Make  their  absolute  terms  alike,  and  subtract.  TJie  result- 
ing equation  has  every  term  of  the  second  degree  in  x  and  y . 


QUADRATIC   EQUATIONS  437 
EXAMPLE 

Solve:                                        ^x^  +  xy^m,  (i) 

I  x2  -  2/2  =  11.  {2) 

Multiplying  {2)  by  6,                       6  a!^  —  6  ?/2  =  66.  (5) 

Subtracting  (i)  from  (5),    &  X^  —  Xy  —  6  y^  =  0.  (4) 

Factoring  (A),                   (5  iB  —  6  y)  (x  +  y )  =  0.  (5) 

Expressing  x  in  terms  of  2/,     OJ  =  f  y  and  x  =—y.  (6) 

Substituting  X  =  f  2/  in  (f),               |f  y^  _  y2  _  H.  (7) 

Solving  (7),                                                          2/ =±5.  (/) 

Substituting  y  =  ±  5  in  x  =  f  2/,                        OJ  =  ±  6.  (5) 

Similarly,  substituting  x  =  -  y  in  (2),  y^  —  y'^  =  11.  (^10) 

But  tbis  leads  to                                                0  =  11.  (ll) 
(11)  being  impossible,  tbe  solution  is 

x  =  ±6,  y  =  ±5.  {12) 

Test.     Taking    the    values  f  (_j_  6)2  _i_(_t.  6)  (±  5)  =  66. 

to  be  both  positive -^         r^fi^2    (^^\1-^^ 

or  both  negative,      {  C±  o;^-C±  o;^  _  11. 

2.   Substitute  vx  /or  y  throughout  the  equations  and  solve  for  v. 


Solve : 

■2aj2-3aJ2/  +  2/2=4, 

(^) 

■ 

a;2  _  2  a;?/ +  3  2/2  =  9. 

■(^) 

Putting  y  =  »x  in  (I)  and  {2) 

2  x2  -  3  vx2  +  t)2a;2  _  4^ 

(5) 

and, 

X2  _  2  Va;2  +  3  ?j2a;2  =:  9. 

(-#) 

Factoring  (5)  and  (4), 

,x2(2-3v  +  v2)  =  4. 

(5) 

x\l  -  2  u  +  3  v2)  =  9. 

(5) 

Equating  the  values  of  x^  in 

4                        9 

i-(^) 

(5)  and  (6), 

2-3v  +  ij2     i_2t>+3t 

Clearing  (7)  of  fractions, 

3^2  +  19^-14  =  0. 

(5) 

Solving  (S), 

^  =  -7,|. 

(^) 

Since  y  =  rx. 

2/  =  -  7  x, 

{10) 

and, 

-T- 

ill) 

From  (1),                    2  JC2  + 

21x2  +  49x2  =  72x2  =  4. 

{12) 

Solving  (i2), 

3V2 

y             7 

{IS) 

From  (10), 

3\/2 

{U) 

Similarly,  from  (jfl)  and  (I), 

x  =  ±3, 

{15) 

From  (i5)  and  (11), 

y  =  ±2. 

{16) 

Test  as  usual. 

438  A   HIGH   SCHOOL  ALGEBRA 


WRITTEN     EXERCISES 

Solve: 

xy  =  6,  Sx'^  —  xy  =  1. 

2.  x'^  —  xy  =  o4:,  S.    x"^ -\- xy  —  2  y^  —  4:f 
xy-y^  =  IS.  x^-3xy  +  2  =  0. 

3.  x^-{-xy  =  12,  9.    x^-\-6xy  +  2y^  =  lS3j 
2/2  +  xy  =  24.  a;2  _  y2  ^  i^ 

4.  4^2  +  3  2/2  =  43,  10.   2a;2  +  3a:2/  +  2/^  =  80, 
3x2-2/2  =  3.  a;2/  +  a''^  =  6. 

5.  ^±1  +  ^  =  5  n.  «^-42/-20, 
a?  — 2/ic  +  2/4  _  J^  ' 
a^  +  2/2  =  20.                                       a;2/  =  lA 

6.  a;2  +  2/2  =  41,  12.   4  a;2  +  3  2/^  =  43, 
a;2/  =  20.  3  a;2  -  2/2  -  3  =  0. 

583.  The  substitution  of  a  single  variable  for  a  function  is 
the  most  successful  means  of  simplifying  quadratic  systems 
that  do  not  yield  to  the  methods  already  given. 


3 

EXAMPLE 

t-y  -y/x-y  =  2, 

x3-y8  =  2044. 

Put 

y/x  —  y  =  z. 

From  (i), 

z'^-z- 2  =  0. 
(z-2)(^  +  l)=0. 

z  =  2,  or  -  1. 

.-.  from  (^), 

x-y  =  4:,  otI. 

a) 

{2) 
(5) 

{5) 
{6) 
From  {2), 

(X  -  y)  (x2  +  x!/  +  ?/2)  =  2044.  (7) 

From  (7)  and  (6), 

a:2  +  x?/  +  y2  =  2044,  or  511.  (<5) 

From  (6),  ?/  =  x-l.     Put  this  in  (5). 

Then,     x2  +  x(x  -  1) +  (x  -  1)2=  2044.  (5) 

x2  +  x2-x  +  ic2-2x  +  l=  2044.  {10) 

a;2  _  X  -  681  =  0.  (ii) 

solving  («),      .=  i±4^a„d,  =  ^^A±^^- 


QUADRATIC   EQUATIONS  439 

From  (6),y  =  x-i.    Put  this  in  (<§).  ' 

a;2_,.a;(x-4)  +  (x-4)2  =  611.  (13) 

3  ic2  -  12  X  -  495  =  0.  iU) 

x2-4x-  165  =  0.  (J5) 


_4± 

>/l6  4-  4.165 

2 

(ig) 

2\/4  +  165 
9. 

(i7) 

=  2  ±13 

(18) 

=  15,  ( 

)r  -  11. 

(19) 

.• 

.2/-ll,< 

3T  -  15. 

(SO) 

Te8' 

r  by  substitution. 

WRITTEN    EXERCISES 

Solve : 

1. 

11. 

a;2  +  2/2  =  25, 
07  +  2/  =  1. 

x  —  y     x  +  y     2' 

12. 

o;-y  =  2, 

2. 

oj2y_o;2/'  =  30. 

a;2  -f-  2/2  =  90. 

13. 

0.-2  +  2/' +2(0;+ y)=  12, 

3. 

a.2^_aj2/  +  2/2  =  7, 

xy—{x-i-y)=2. 

aj4  _^  ^2^,2  +  a;4  =  21. 

14. 

0^2  4-4  2/2  =  13, 

4. 

aj2  _^  2/2  —  1  =  2  iBi/, 

0;  +  2  2/  =  5. 

aj?/(a;?/  +  l)  =  8190. 

15. 

2a;  4- 2/=  5, 

5. 

a^2+a;2/=f2, 

3o;2-72/2  =  5. 

a^^  +  2/'  =  A- 

16. 

x-y  =  5,_      ■ 

6. 

x^-^2xy  —  55, 

V^  —  V2/  =  1. 

2x2-a;?/  =  35. 

17. 

o;2  +  .'i^_+2/'  =  21, 

7. 

2  0^2  4.  ^2/ =  24, 

a;  _  ^xy  +  2/  =  3. 

a;2_^2  =  5. 

Suggestion.     In  Ex.    17 

divide 

*"      ^ 

(1)  by  (2),  obtaming 

8. 

x-\-y  +y/xy  =  7. 

9. 

0^2-2/2  =  45, 

Add  this  equation  to    (2) 
find  X  in  terms  of  y. 

and 

xy-hy^=  18. 

10. 

05-22/--=  2, 

a;2  _  6  2/2  =  14  —  071/. 

18. 

^2  4-  te  +  «'  =  133, 

i  +  0;  —  V^  =  7. 

440  A  HIGH   SCHOOL   ALGEBRA 

19.  x^-{-2xy-\-7y''  =  24:,  22.    (2  x -S)(3y  -  2)  =  0, 
2x^-xy-y^=S.  4  a:^  -\-  12xy-3y^  =  0. 

20.  x(x  —  y)=0,  Suggestion.      In    Ex.  22,  from 
x'^+2xy-^y^  =  9,  equation  (1),    x  =  |.    The  cor- 
responding value  of  y  is  found 
by  substituting  this  value  of  x 


21.   Vx  —  Vy  =  2, 


(Va;  —  ■Vy)'\/xy  =  30.  in  equation  (2). 

23.  Two  men,  A  and  B,  dig  a  trench,  in  20  days.  It  would 
take  A  alone  9  days  longer  to  dig  it  than  it  would  B.  How 
long  would  it  take  A  and  B  each  working  alone? 

Note.  Those  who  have  studied  geometry  and  are  ready  for  problems 
based  upon  geometric  properties  will  find  them  in  Chapter  XXXIII. 


REVIEW 

WRITTEN     EXERCISES 
Solve: 

1.  x^=6x-5.  17.  x'^  =  6x  +  16. 

2.  w^-w  — 1  =  0.  18.    24 -  10 a;  =  x\ 

3.  a.2_6a,_7==o.  19.    (x-iy  =  x  +  2, 
4.-^2+2^  +  6  =  0.  20.  5ic  +  a;2  +  6  =  0. 

5.  x'^-5x-\-l=0.  21.   a^-9x  +  U  =  0. 

6.  2i»2  — fl;  +  3  =  0.  22.    a;2+ 3  0^-70  =  0. 

7.  3a;2_a;  +  7  =  0.  23.   4a;2_  4  a;-3  =  0. 

8.  a.'2-5a;  +  ll  =  0.  24.    3  x^- 7  ic  +  2  =  0. 

9.  x~^Ux  +  5=0.  25.   a;2_10a;  +  21=0. 

10.  3aj2-9a;-|=0.  26.   ic^-lOa; +  24=  0. 

11.  »2_4a;_45  =  0.  27.   a;2  + iOa.  + 24  =  0. 

12.  a;2_4a.  +  45:^0.  28.   x"^  =  9  x'^  —  (x  +  ly. 

13.  a;2  +  4a;-45  =  0.  29.   9  a;2  +  4a;- 93  =  0. 

14.  a;2  +  4£c  +  45=0.  30.   4a;2  +  3a;-22  =  0. 

15.  a;2  +  30a;  +  221=0.  31.    6  a;^- 13  a;  +  6  =  0- 

16.  a;2-30a;-221  =  0.  32.    (ic  + l)(a;  +  2)  =  x  +  3. 


QUADRATIC   EQUATI0:NS  441 


a^     a? 


33.  aj  +  -  =  — +&•  36. V- =  C. 

X       h  0  -^x      0  —  X 

34.  cx''-hbx-\-a  =  0.  x  +  1^        a;  + 12    ^1 

35.  (x-2)(x-{-S)  =  16.  '     x-\-2       2(a;  +  19)      2* 

38.  5.-5(^  =  2.4-?^^^. 

.-3  2 

39.  (x-l)Xx-hS)  =  x(x+5)(x-2). 

40.  .2  _  6  acx  +  ci2(9  c2  -  4  6^)  =  0. 

41.  1  +  1  +  1 1-^  =  0. 

abxa-\-b-\-x 

42.  l  +  ^  +  ^_  =  0. 

43.  (2  a  -  5  -  .)2+  (3  a  -  3  .)2=(a  +  5-2  .)2. 

44.  (3  .-  4  a  +  3  by +(2  x  +  a)2=(a;  -  4  a  +  6)^ 

+  (2.-3a4-4&)2. 

45.  (7  a  +  3  6  +  .)2+(4  a  -  6  -  8  .)2_(4  a  +  3  &  +  4  .)2 

=  {1  a+b-^xy. 

46.  (.  +  a)(5a;-3a-46)  =  (.-+-a-2&)2. 

47.  (5  a;  +  4  a  +  3  6)  (10  a;  -  6  a  +  8  6)  =  (5  .  +  a  4-  7  by. 

48.  (26a;4-a  +  22  6)(14a;  +  13a-2  6)  =  (16.  +  lla  +  8&)2. 

49.  (8  c  + 10  4-  4  .)  (18  c  4- 160  -h  24  a;)  =  (12  c  4-  40  4-  H  xy. 
14?/2-hl6       2/4-8  ^2y2 

21  8  2/2-11        3   ' 

a2  (6-c)2' 

.2.^3    +    ^,2  _^  9 

g3      .24-. -2       --«  +  l^Q 
•^2_^2.-3      .-3 

^A  1  1,1      1 

54.  — =-  +  7 

a-\-b  —  X     a      b      X 

55.  (a-x){l-^-^±^\-2=\^{c-^)       ^+^' 


50. 


2c 


442  A  HIGH   SCHOOL   ALGEBRA 


56. 

a;2  H-  cc?/  +  4  2/2  =  6, 
3x''  +  Sy^  =  14:. 

62. 

X^-y^  =  112, 

x-\-y  =  U. 

57. 

a^  -f  2/3  =  ctj 
x-^y  =  b. 

63. 

«  +  2/  =  7, 
0^3  +  2^^133. 

58. 

x^2y  =  2i. 

64. 

x^-xy  +  y^  =  61. 

59. 

x^-\-xy  =  S6, 

65. 

o?-f  =  m^, 

X^-y1  =  -^, 

x—  y  =5. 

60. 

x'  +  ^xy=^22, 
x  +  y  =  ^. 

66. 

x'^y  +  ic/  =  12. 

61. 

0^2  +  42/2=85, 
x-y^2. 

67. 

0,2  _  ^y  +  2/2=  37, 

68. 

X          ,            5 

24 

x'+x  +  b      Vic2  +  a 

;  +  5      4a; 

S 

iuGGESTiON.     Multiply  bo 

th  members  by  x  and  let 

V-           ^ 

69. 

Vx^  +  a;  +  5 

State  what  can  be  known  by  means  of  the  discriminant  and 
without  solving,  concerning  the  factors  of : 

70.  3x2_2a;  +  l.  72.   52/^  +  202/ +  20. 

71.  4o;2  +  lla;  — 1.  73.    2o;2— o;  +  3. 
Similarly,  what  can  be  known  about  the  roots  of : 

74.  ^2_9^()9  76.     ;22_2_1  ^Q9 

75.  5o:2  +  o;+2  =  0?  77.    5o;2=9i»? 

How  must  a  be  chosen  in  order  that : 

78.  The  roots  of  o;2  -f-  ao;  +  5  =  0  shall  be  imaginary  ? 

79.  The  roots  of  ax2  +  6  o;  +  1  =  0  shall  be  real  ? 

80.  The  roots  ofo;2  +  4o;  +  2a  =  0  shall  be  real  and  of  oppo- 
site signs  ? 

81.  The  roots  of  (a  +  1)  a;2  -f-  3  o;  —  2  =  0  shall  be  imaginary  ? 

82.  The  roots  of  4  o;2  —  ao;  +  2  =  0  shall  be  real   and  both 
positive  ? 


QUADRATIC   EQUATIONS  443 

Find  the  values  of  m  for  which  the  roots  of  the  following 
equations  are  equal  to  each  other.  What  are  the  correspond- 
ing values  of  a;  ? 

83.  x^  —  12  X  -\-  S  m  =  0.  85.    4:x'^-\-mx-^x -{-1  =  0. 

84.  7nx'^ -{- S  X -\- m  =  0.  86.   mx'^  i- S  mx  —5  =  0. 

87.  A  number  increased  by  30  is  12  less  than  its  square. 
Find  the  number.  ' 

88.  The  product  of  two  consecutive  odd  numbers  is  99. 
What  are  the  numbers  ?     Is  there  more  than  one  set  ? 

89.  Find  two  consecutive  even  numbers  the  sum  of  whose 
squares  is  164. 

90.  Find  a  positive  fraction  such  that  its  square  added  to 
the  fraction  itself  makes  -f. 

91.  If  a  denotes  the  area  of  a  rectangle  and  p  its  perimeter, 
show  that  the  lengths  of  the  sides  are  the  roots  of  the  equation 

x^-Ex-{-a  =  0. 

92.  The  diagonal  and  the  longer  side  of  a  rectangle  are  to- 
gether equal  to  5  times  the  shorter  side,  and  the  longer  side 
exceeds  the  shorter  by  35  meters.  Find  the  area  of  the 
rectangle. 

93.  A  company  of  soldiers  attempts  to  form  in  a  solid 
square,  and  5^  are  left  over.  They  attempt  to  form  in  a 
square  with  3  more  on  each  side,  and  there  are  25  too  few. 
How  many  soldiers  are  there  ? 

94.  It  took  a  number, of  men  as  many  days  to  dig  a  ditch  as 
there  were  men.  If  there  had  been  6  more  men,  the  work 
would  have  been  done  in  8  days.     How  many  men  were  there  ? 

95.  Solve:  2x^^Q)X^c  =  0. 

What  value  must  c  have  to  make  the  two  values  of  x  equal  ? 

96.  A  library  spends  $  180  monthly  for  books.  In  June  the 
average  cost  per  book  was  15  ^  less  than  in  May,  and  60  books 
more  were  bought.     How  many  were  bought  in  May  ? 


444  A  HIGH   SCHOOL   ALGEBRA 

97.  When  water  flows  from  an  orifice  in  a  tank  the  square 
of  the  velocity  (v)  equals  2  g  times  the  height 
(h)  of  the  surface  above  the  orifice.  Write 
the  equation  that  denotes  this  fact,  g  is 
the  "  constant  of  gravity  ''  and  may  be  taken 
as  32. 

98.    What  does  the  square  of  the  velocity, 
(v^),  at  A  in  the  figure  equal  ?     Eind  this  velocity. 

99.  What  would  be  the  velocity  of  the  water  if  an  opening 
were  made  halfway  up  from  A  shown  in  the  figure  ? 

100.  Find  the  velocity  with  which  water  rushes  through  an 
opening  at  the  base  of  a  dam  against  which  the  water  stands 
25  ft.  high. 

SUMMARY 

The  following  questions  summarize  the  definitions  and  pro- 
cesses treated  in  this  chapter  : 

1.  State  the  general  forms  of  the  roots  of  the  general  quad- 
ratic equation.  Sec.  563. 

2.  Statetherelation  betweenthe  roo^s  and  coe^cie?ife.  Sec.  566. 

3.  Name  two  symmetric  functions  of  the  roots  of  the  quad- 
ratic equation.  Sec.  567. 

4.  What  is  the  discriminant  of  the  quadratic  ?         Sec.  572. 

5.  What  is  the  nature  of  the  discriminant  6^  —  4  ac  when 
the  roots  are : 

1.  Real  and  unequal  ? 

2.  Eeal  and  equal  ? 

3.  Complex  numbers? 

4.  Irrational  numbers  ?  Sec.  571. 

6.  What  methods  are  used  to  solve  radical  equations  ? 

Sees.  576-579. 

7.  What  kind  of  higher  equations  may  be  solved  like 
quadratics  ?  Sec.  580. 

8.  What  kind  of  equations  does  x"  ±a=0  represent ? 

Sec.  581. 


QUADRATIC   EQUATIONS 


445 


9.    State  the  three  cube  roots  of  unity  in  terms  of  1  and  w. 

Sec.  581. 

10.    What  method  of   solving  simultaneous   quadratics   has 

the  broadest  application?  Sec.  583. 


HISTORICAL  NOTE 

Among  the  many  improvements  in  algebra  made  in  the  sixteenth 
century  was  that  masterly  achievement,  the  general  solution  of  the  cubic 
equation.  We  have  explained  how  for  centuries  the  solution  of  the  quad- 
ratic equ?itionjc^-^'-px.j=^  baffled  the  skill  of  the  keenest  mathematicians, 
but  the  solution  of  the  cubic  x^  +  px^  =  q  had  been  regarded  quite 
unattainable,  so  much  so  that  the  great  Italian  Paciuolo  (1494)  closed  his 
famous  work  Summa  with  the  remark  that  the  solution  of  this  equation 
was  as  impossible  as  the  squaring  of  the  circle.  But  to  assert  that  the 
solution  of  a  problem  is  impossible  is  dangerous  ground  to  take,  as  was 
proved  in  this  case.  For  within  fifty  years  after  this  remark  of  Paciuolo, 
his  countryman  Niccolo  solved  the  cubic  equation. 

Niccolo  was  born  at  Brescia  just  at  the  beginning  of  the  sixteenth 
century,  and  came  to  be  known  asTartaglia,  meaning,  "  The  stammerer." 
This  defect  in  his  speech  was 
caused  by  a  saber  cut  received 
at  the  hands  of  a  French  sol- 
dier when  Niccolo  was  six 
years  old.  Although  poverty 
deprived  him  of  the  advan- 
tages of  school  instruction, 
his  industry  and  persistence 
enabled  him  to  master  the 
classics  and  mathematics,  and 
to  add  (1541)  to  the  science 
of  algebra  the  most  significant 
discovery  of  the  sixteenth 
century.  It  was  Tartaglia's 
ambition  to  write  a  great  work 
on  algebra  and  to  use  this  as 
a  means  of  giving  to  the  world 
his  method  of  solving  the  cubic 
equation.  But  before  he  could 
accomplish  this,  he  was  be- 
trayed  by  his  friend  Cardan, 

to  whom,  after  mucti  pTeaHmg,  he  had  revealed  the  nature  of  his  dis- 
covery.    Cardan  published  his  now  famous  work,  the  Ars  Magna,  in  1545, 


Niccolo,  or  Tartaglia 


446         A  HIGH  SCHOOL  ALGEBRA 

and  included,  as  the  crown  to  his  treatment  of  equations,  the  solution  by 
Tartaglia,  claiming  for  himself  full  credit  for  the  discovery.  This  decep- 
tion and  robbery  was  a  bitter  disappointment  to  Tartaglia,  who  never 
recovered  sufficiently  to  complete  his  projected  writings,  and,  although 
the  claim  of  Cardan  is  now  conceded  by  all  to  be  fraudulent,  the  solution 
of  the  bubic  equation  is  still  commonly  called  Cardan's  method. 

The  solution  of  the  equation  of  the  fourth  degree  immediately  followed 
that  of  the  cubic  equation.  This  was  first  effected,  in  a  similar  way,  by 
Ferrari,  a  pupil  of  Cardan,  and  was  also  published  in  the  Ars  Magna. 
It  was  naturally  supposed  that  these  methods  could  be  extended  to  equa- 
tions of  degrees  higher  than  the  fourth,  and  prodigious  labor  was  expended 
in  the  effort  to  do  this,  until  Abel,  a  Norwegian  mathematician,  proved 
in  1824  that  the  methods  of  elementary  algebra  are  not  sufficient  to  solve 
general  equations  of  degree  higher  than  the  fourth. 


CHAPTER  XXXI 

PROPORTION^,  VARIATION,   AND  LIMITS 
PROPORTION 

584.  Proportion.     If  four  numbers,  a,  b,  c,  d,  are  such  that 

-=  -,  they  are  said  to  form  the  proportion,  a  is  to  b  sls  c  is  to  d. 
h     d 

585.  Fourth  Proportional.  If  four  numbers  a,  h,  c,  d  are  in 
proportion,  d  is  called  the  fourth  proportional  to  a,  6,  and  c. 

586.  Third  Proportional.  If  three  numbers,  a,  6,  c,  are  such 
that  a,  6,  6,  c  are  in  proportion,  c  is  called  the  third  proportional 
to  a  and  6. 

587.  Mean  Proportional.  If  a,  6,  6,  c  are  in  proportion,  6  is 
called  the  mean  proportional  to  a  and  c. 

588.  Means  and  Extremes.  If  a,  6,  c,  d  are  in  proportion, 
h  and  c  are  called  the  means,  and  a  and  d  the  extremes. 

589.  Relation  of  Means  to  Extremes.  In  any  proportion  the 
product  of  the  means  equals  the  product  of  the  extremes,  and  con- 
versely, if  the  product  of  two  numbers  equals  the  product  of  two 
other  numbers,  the  four  numbers  can  be  arranged  in  a  proportion. 

For,  the  two  factors  of  one  product  may  be  made  the  means,  and  the 
two  factors  of  the  other  the  extremes,  or  vice  versa. 

590.  Inversion.     If  -  =  -,  then  ^  =  ^. 

b      d  a      c 

For,  the  members  of  the  second  equation  are  the  reciprocals  of  the 
members  of  the  first, 

591.  Alternation.     If  ^  =  ^,  then  ^  =  ^. 

b      d  c      d 

For,  the  second  equation  results  if  the  first  is  multiplied  by  -. 

c 
447 


448  A   HIGH  SCHOOL   ALGEBRA 

592.  Composition.     If  ^  =  ^,  then  «  +  ?>^c+j 

For,  the  second  equation  results  from  the  first  by  adding  1  to  both 
members. 

593.  Division.     If  ^  =  -^  then  ^-IH^^^:^. 

b     d'  b  d 

For,  the  second  equation  results  from  the  first  if  1  be  subtracted  from 
both  members. 

594.  Composition  and  Division.   If  -  =  - ,  then  ^^±J  =  ^  +  ^ . 

b      d  a—b     c—d 

For,  the  result  of  Sec.  592  divided  by  the  result  of  Sec.  593  gives  the 
second  equation. 

595.  Continued  Proportion.     If  several  ratios  are  equal,  as  in 

7  =  -,  =  ^ . . .  the  numbers,  a,  6,  c,  d,  e,  /, . . . ,  are  said  to  be  in 
b      d     f 

continued  proportion. 

In  the  continued  proportion   -  =  -  =  ^  . . . ,  any  one  of  the 

a  +  c  +  e...  b     d     f 

ratios  equals  --^ — — . 

596.  These  properties  may  be  applied  to  certain  problems. 

EXAMPLES 

1.  If  p  and  w  are  the  power  and  weight  applied  to  a  lever 
whose  power  and  weight  arms  are,  respectively,  a  and  b,  then 
pa  —  bw.     Write  a  proportion  between  these  four  numbers. 

Solution.    By  Sec.  589,  a:b  =  w.p. 

2.  Find  the  third  proportional  to  a  —  6  and  a^  —  W. 

SoLLTiON.     1.   By  Sec.  689,    «-^  =  ^'"^  -  ^^ 

2.   .-.  (a-6)a;  =  (a2_62)(aj2_52). 
'   3.    .  •.  the  third  proportional,  oj  =  (a  +  &)(a2  —  62). 

3.  Form  a  proportion  from  the  equation,  x^  —  y'^  =  z\ 

Solution.     1.   Factoring,  {x  —  y)(x  +  y)  =  zz. 

2.    .-.  ^IlJ^  =  — ^— .     Sec.  589. 
z         x-\-y 


PROPORTION,   VARIATION,   AND  LIMITS 


riven  -  =  - ,  show  that 
b      d 

a  -f  6      Vo^  +  b^ 

By  Sec.  592 

a-i-  b  _c  +  d 
b            d 

.-.  by  Sec.  591, 

a  +  b  _b 
c  +  d      d 

Squaring  both  members  of  the 
given  equation, 

6-2       d^' 

Applying  Sec.  592  to  (4), 

a2,+  62       c2  +  ^2 
62                     d2 

Applying  Sec.  591  to  (5), 

aP-  +  &2      &2 

C^  +  fZ2        d2 

From  (6), 

Va2  4-  6-2  _  6 

Vc2  +  d2      f^ 

Equating  values  of  -  in  (3)  and 

a  +  />  _  Va2  +  6'^ 

c  +  d 


d^ 


449 

(5) 

(7) 


WRITTEN    EXERCISES 

1.  A  lever  need  not  be  straight,  although  it    ^. 
must  be  rigid.     Thus,  the  crank  and  the  wheel  and     "^ 
axle  are  varieties  of  the  lever,  and  the  law  of  the 
lever  (p.  188)  applies  to  them.     Thus,  in  the  figure, 

W     p' 
Find  W,  if  P  =  14,  p  =  16,  and  iv  =  4. 

2.  Find  the  unknown  number  : 


(1) 

(2) 

(3) 

p= 

3a 



a-b 

w  = 

66 

5p 

(a  +  bf 

P=: 

2  c 

Sp 

a-{-b 

IV  = 

— 

2p 

— 

450 


A  HIGH   SCHOOL   ALGEBRA 


3.  If  an  axle  is  6  in.  in  diameter,  what  must  be  the 
diameter  of  the  wheel  in  order  that  a  boy  exerting  a  force  of 
50  lb.  may  be  able  to  raise  800  lb.  weight  ? 

4.  A  brakeman  pulls  with  a  force  of  150  lb. 
j  on  a  brake  wheel  16  in.  in  diameter.     The  force 

is  communicated  to  the  brake  by  means  of  an 
axle,  A,  4  in.  in  diameter.  What  is  the  pull  on  the 
brake  chain  ? 

5.  In  the  figure  below  the  weight  W  acts  at  P  on  an 
inclined  plane,  whose  rate  of  slope  is  a  vertical  units  to  b 
horizontal  units.  It  is  known  that  the  weight  W  acts  in 
two  ways  :  a  force  TV  pressing  directly  against  the  surface 
and  tending  to  produce  friction,  and  a  force  F  parallel  to 
the  plane  and  tending  to  cause  ^ 

the  weight  to  slide  down  the  plane.  It  is 
known  that  these  various  quantities  are  re- 
lated to  each  other  thus : 

W      l'  W      l' 


Eind  F  and  ^,  if  W=  15  lb.,  a  =  20  in.,  6  ==  21  in.,  1=29  in. 

6.  Find  I  and  TV,  if  F=  16  lb.,    W=  34  lb.,  a  =  4  ft.   5  = 
71ft. 

7.  Find   b  and  I,  if    F=66lh.,  iV=1121b.,   TF=1301b., 
a  =  33  in. 

8.  Find  TTand  a,iiF=  200  lb.,  7V=  45  lb.,  b  =  9,  I  =  4.1. 

9.    It  is  known  that  in  any  cone  the  area  of 

any  section  parallel  to  the  base  varies  as  the 

^    square  of  its  distance  from  the  vertex.     Thus,  in 

the  figure,  -  =  — • 
s       d^ 

li  d  =  1,  d'  =  ^,  and  s  =  40  sq.  in.,  what  is  the  area  of  s'  ? 

10.    The  area  of  a  section  ^  of  the  way  from  the  vertex  to 
the  base  and  parallel  to  it  is  what  part  of  the  base  ? 


-.ss^^^ffT 

b 

'i  > 

PROPORTION,   VARIATION,   AND   LIMITS  451 

11.  Given -  =  -,  show  that  — —  = ■. 

be  a-{-c        a  —  c 

12.  Given  -  =  -,  show  that  ciP-^^b^^-_l^ 

be  a  c 

13.  Given  -  =  -,  show  that 


b      d'  "        d{a-\-by     b{d-\-cy 

14.    If  a  :  6  =p  :  q,  prove  that 


a2  4-62.       «'      -p2_|_^2.      i^'      ^ 

a  +  6                   p  +  g 

Suggestion. 

The  given  proportion  may  be  written 

a_b^ 
P     a' 

Let 

-  =  r,  then  a  =pr,  h  =  or. 
P 

The  proportion  to  be  proved  may  be  written  : 

(a  +  b)(a^  +  h^)  ^  (P  +  q)(p^  +  q^)  ^ 

Substituting  the  values  of  a  and  b  above,  the  left  member  readily 
reduces  to  the  right. 

Note.  A  good  method  for  proving  such  identities  is  to  begin  with  the 
required  relation  and  transform  it  into  the  given  relation,  or  to  transform 
both  the  given  and  the  required  relation  until  they  reduce  to  the  same 
thing. 

15.  Given  ?=  «   show  that  ii^=l*?  =  i^!-^!^. 

b      d  o¥  bd? 

ft  ft  q2  7j2  7^2 

16.  Given  -  =  -,  show  that  — -  =  —• 

b      d  c'-d^      d"" 

17.  Write  oc^  —  4:  y"^  =  x^  —  xy  in  the  form  of  a  proportion. 


18.   Given  «  =  ^,  show  that  «  =  X0^. 
b     d  c      ^d'  —  d' 


452  A  HIGH   SCHOOL   ALGEBRA 

VARIATION 

597.  Direct  Variation.  AVhen  two  variable  quantities  vary 
so  as  always  to  remain  in  the  same  ratio,  each  is  said  to  vary 
directly  as  the  other.  Each  increases  or  decreases  at  the  same 
rate  that  the  other  increases  or  decreases. 

Consequently,  if  x  and  y  are  two  corresponding  values  of  the 
variables  and  k  the  fixed  ratio,  then, 

-  =  k  and  x  =  ky. 

y 

For  example,  at  a  fixed  price  {k)  per  article,  the  total  cost  (x)  of  a 
number  of  articles  of  the  same  sort  varies  directly  with  the  number  of 
articles  {y) .     That  is  -  =  A;. 

y 

Likewise  in  the  case  of  motion  at  a  uniform  rate  (r) ,  the  distance  tra- 
versed {d)  varies  as  the  time  of  motion  (J) .     That  is,  -  =  r. 

598.  A  symbol  still  occasionally  used  for  "  varies  as  "  is  oc. 

Thus,  "  X  varies  as  y  "  is  written  xccy, 
and  "  d  varies  as  « "  is  written  dcct. 

599.  Relation  of  Variation  to  Proportion.  When  one  vari- 
able varies  directly  as  another,  any  pair  of  values  of  the  vari- 
ables forms  a  proportion  with  any  other  pair. 

x'  x"  x'     x" 

For,  —  =  r,  and  —-  =  ri     .•.—  =  -—  which  is  a  proportion. 

y  y  y'    y 

600.  Expressions  for  Direct  Variation.  We  have  thus  seen 
that  the  relation  x  varies  directly  as  y  may  be  expressed  in  any 
one  of  three  ways : 

(a)  X  =  ky,  by  use  of  the  equation. 

(6)  xccy,  by  use  of  the  symbol  of  variation. 

x'     x" 
(c)    —  =  -—  by  use  of  the  proportion;  x',  y'  and  x",  y"  being 

y     y 

any  two  pairs  of  corresponding  values  of  the  variables. 


WRITTEN    EXERCISES 

1.   Write  the  statement  "  v  varies  as  iv  "  in  the  form  of  an 
equation,  also  in  the  form  of  a  proportion. 


PROPORTION,  VARIATION,   AND   LIMITS  453 

2.  Write  the  statement  x  =  ky  by  use  of  the  symbol  oc ; 
also  in  the  form  of  a  proportiom. 

3.  Write  —  =  — r  by  use  of  the  symbol  oc. 

4.  The  weight  {w)  of  a  substance  varies  as  the  volume  (v) 
when  other  conditions  are  unchanged.  Express  this  law  by 
use  of  the  equation.  By  use  of  the  symbol  oc.  Also  in  the 
form  of  a  proportion. 

5.  In  the  equation  w  =  kv,  if  w;  =  4  and  v  =  2,  what  is  the 
value  of  A:?  Using  this  value  of  k,  what  is  the  value  of  w 
when  v  =  25? 

6.  When  1728  cu.  in.  of  a  substance  weigh  1000  ounces, 
what  is  the  ratio  of  the  weight  (w)  to  the  volume  (v)  ?  What 
volume  of  this  substance  will  weigh  5250  ounces  ? 

7.  The  cost  (c)  of  a  grade  of  silk  varies  as  the  number  of 
yards  (n).     Find  the  ratio  (r)  of  c  to  n  when  c  is '$7.00  and 

71  =  4. 

c  c' 

8.  In  Exercise  7  if  —  =  r,  what  does  —  equal  ?     Given  that 

n  ...  4-,.,  n 

40  yd.  of  silk  cost  $  60,  find  the  cost  of  95  yd.  by  means  of  the 

proportion  —  =  —     Also  by  means  of  the  equation  c  =  nr. 

601.  Inverse  Variation.  A  variable  x  is  said  to  vary  inversely 
as  a  variable  y,  if  it  varies  directly  as  — 

Inverse  variation  means  that  when  one  variable  is  doubled  the  other  is 
halved ;  when  one  is  trebled  the  other  becomes  ^  of  its  original  value, 
and  so  on. 

602.  Expressions  for  Inverse  Variation.  The  relation  "a; 
varies  inversely  as  y  "  may  be  expressed  : 

(1\  k 

-  J,  or  a;  =  —    .*.  xy  =  k. 

(b)  With  the  symbol  of  variation  y  oc  -• 

X 

(c)  As  a  proportion  —  =  ^^ . 

X        y 


454  A    HIGH   SCHOOL   ALGEBRA 


WRITTEN     EXERCISES 

1.  Write  the  statement  "  v  varies  inversely  as  w "  in  the 
form  of  an  equation ;  also  in  the  form  of  a  proportion. 

2.  Write  ^  =  -  by  use  of  the  symbol  oc ;  also  in  the  form  of 

a  proportion. 

3.  In  a  bicycle  pump  the  volume  (v)  of  air  confined  varies 
inversely  as  the  pressure  {p)  on  the  piston.  Write  the  rela- 
tion between  v  and  j?  in  three  ways. 

4.  In  Exercise  3,  if  v  =  18  (cu.  in.)  and  j9  =  15  (lb.),  what 

is  A;  in  V  =  - ?     What  is  the  pressure  (p)  when  v  =  1  (cu.  in.)? 
P 

5.  In  an  auditorium  whose  volume  (y)  is  25,000  cu.  ft. 
there  are  2000  persons  {p).  What  is  the  number  (n)  of  cubic 
feet  of  air  space  to  the  person?  What  will  be  the  number 
when  1000  more  persons  come  in  ? 

6.  The  area  of  a  triangle  varies  as  the  base  times  the  alti^ 
tude.  If  the  area  is  12  when  the  base  is  8  and  the  altitude  3, 
what  is  the  area  of  a  triangle  whose  base  is  40  and  altitude  20  ? 

7.  The  area  of  a  circle  varies  as  the  square  of  its  radius. 
The  area  of  a  circle  of  radius  2  is  12.5664 ;  what  is  the  area  of 
a  circle  whose  radius  is  5.5  ? 

8.  The  volume  of  a  sphere  varies  as  the  cube  of  its  radius. 
If  the  volume  of  a  sphere  whose  radius  is  3  is  113.0976,  what 
is  the  volume  of  a  sphere  whose  radius  is  5  ? 

9.  If  a;  oc  2/  and  cc  =  6  when  y  =  2,  find  x  when  y  =  8. 

Suggestion.     x  =  ky.     .-.6  =  k  -  2  and  k  =  S.    Substitute  y  =  S  in  the 
equation  x  =  Sy. 

10.  Determine  k  in  xccy,  if  a;  =  10  when  y  =  20.     Also  if 
a?  =  1  when  2/  =  5.     If  x  =  100  when  y  =  10. 

11.  li  xccw  and  y  ccw,  prove  that  x-\-y  ccw. 

12.  li  xccw  and  wocy,  prove  that  xy  oc  w^. 

X     ?y 

13.  If  a;  oc  V  and  w  ccz,  prove  that  —  oc  -  • 

w     z 


PROPORTION",   VARIATION,  AND  LIMITS 


455 


14.  If  ic  oc  'ifz  and  a;  =  1  when  2/  =  2  and  2j  =  3,  find  the  con- 
stant A:. 

15.  Given  y  —  z-\-w,  z^x  and  lo  x  a? ;  and  that  cc  =  1  when 
w  =  6,  and  that  x—2  when  z  =  20.     Express  y  in  terms  of  x. 

Suggestion,     y  =  Tcx  +  k'x;  determine  k  and  k' . 

16.  Solve  Exercise  15  under  the  conditions  that  x  =  2  when 
w  =  12,  and  that  a;  =  1  when  z  =  10. 

17.  Given  zacx-{-y  and  y  oc  a^^,  and  that  x  —  ^  when  2/  =  ^ 
and  z  =  ^.     Express  z  in  terms  of  x. 

603.    A  large  number  of  problems  in  science  may  be  solved 
by  the  following  plan  : 


EXAMPLES 

1.  The  "  law  of  gravitation  "  states  that  the  weight  of  a  given 
body  varies  inversely  as  the  square  of  its  distance  from  the 
center  of  the  earth.  What  is  the  weight  of  a  body  5  mi.  above 
the  surface  of  the  earth,  which  weighs  10  lb.  at  the  surface 
(4000  mi.  from  the  center)  ? 

Method.  There  are  two  variables  in  the  problem,  the  weight  («;)  and 
the  distance  (d) .  There  are  also  two  parts  or  cases  in  the  statement,  one 
in  which  the  value  of  one  variable  is  unknown,  and  one  in  which  the 
values  of  both  variables  are  given. 

Arrange  the  data  as  follows  : 


w 

d 

1st  case 
2d  case 

X 

10 

4005 
4000 

To  this  table  apply  the  law  of  variation  expressed  in  the  physical  law. 
Since  the  law  is :  w  varies  inversely  as  the  square  of  c?,  the  values  of  d 
must  be  squared,  and  the  ratio  x  :  10  equals  the  inverse  ratio  of  4005^  to 
40002  .  that  is, 

x^ 
10 
4000  ^  4005 


40002 


80 


40052 
.99874  +  and  .99872  =  9.96  . 
.-.  X  =  9.96,  and  the  weight  is  9.96  lb. 


456  A   HIGH   SCHOOL   ALGEBRA 

2.  The  squares  of  the  times  of  revolution  of  the  planets 
about  the  sun  vary  directly  as  the  cubes  of  their  distances 
from  the  sun.  The  earth  is  93,000,000  mi.  from  the  sun,  and 
makes  a  revolution  in  approximately  365  da.  How  far  is 
Venus  from  the  sun,  its  time  of  revolution  being  226  da.  ? 

Solution* 


t  =  time  of  reyol. 

d  =  distance  from  sun 

1st  case 
2d  case 

365 

226 

93,000,000 

X 

According  to  the  astronomical  law  the  times  must  be  squared  and  the 
distances  cubed  ;  then,  since  the  law  is  that  of  direct  variation  : 

3652  ^  93,000,0008 
2262  x^ 

.-.  a;  =  93,000,000 -Wf^^)    =68,900,000,   and  the  distance  of  Venus 
'  \365/ 

from  the  sun  is  approximately  69,000,000  miles. 


WRITTEN    EXERCISES 

1.  The  intensity  of  light  from  a  given  source  varies  in- 
versely as  the  square  of  the  distance  from  the  source.  If  the 
intensity  (candle  power)  of  an  electric  light  is  4  at  a  distance 
of  150  yd.,  what  is  its  intensity  at  a  distance  of  25  yd.  ? 

2.  According  to  the  first  sentence  of  Exercise  1,  how  much 
farther  from  an  electric  light  miist  a  surface  be  moved  to 
receive  only  \  as  much  light  as  formerly  ? 

3.  The  time  of  oscillation  of  a  pendulum  varies  directly  as 
the  square  root  of  its  length.  What  is  the  length  of  a  pendu- 
lum which  makes  an  oscillation  in  5  sec,  a  2-second  pendulum 
being  156.8  in.  long  ? 

4.  According  to  Exercise  3,  what  is  the  time  of  oscillation 
of  a  pendulum  784  in.  long  ? 

5.  The  distance  through  which  a  body  falls  from  rest  varies 
as  the  square  of  the  time  of  falling.  A  body  falls  from  rest 
576  ft.  in  6  sec. ;  how  far  does  it  fall  in  10  sec.  ? 


PROPORTION,   VARIATION,   AND   LIMITS 


45T 


6.  Volumes  of  similar  solids  vary  as  the  cubes  of  their 
linear  dimensions.  The  volume  of  a  sphere  of  radius  1  in.  is 
4.1888  cu.  in. ;  what  is  the  volume  of  a  sphere  whose  radius  is 
5  in.  ? 

7.  According  to  Exercise  6,  what  is  the  radius  of  a  sphere 
whose  volume  is  33.5104  cu.  ft.  ? 

8.  According  to  Exercise  6,  if  a  flask  holds  ^  pt.,  what  is 
the  capacity  of  a  flask  of  the  same  shape  4  times  as  high  ? 

9.  In  compressing  a  gas  into  a  closed  receptacle,  as  in  pump- 
ing air  into  an  automobile  tire,  the  pressure  varies  inversely 
as  the  volume.  If  the  pressure  is  25  lb.  when  the  volume  is 
125  cu.  in.,  what  is  the  pressure  when  the  volume  is  115  cu.  in.  ? 

10.  According  to  Exercise  9,  if  the  pressure  is  50  lb.  when 
the  volume  is  250  cu.  in.,  what  is  the  volume  when  the  pres- 
sure is  10  lb.  ? 

11.  It  is  known  that  if  one  gear  wheel  turns  another  as  in  the  figure, 
the  number  of  revolutions  of  the  two  R, 
are  to  each  other  inversely  as  their 
number  of  teeth.  That  is,  if  the  first 
has  Oi  teeth  and  makes  Bi  revolutions, 
and  the  second  has  C^  teeth  and  makes 
in  the  same  time  7^2  revolutions, 

Bi^C2 

B2      Ci 
Find  B2  if  Ci=  25,  a  =  15,  and  R,  =  6. 
Find  the  numbers  to  fill  the  blanks  : 


then 


0) 

(2) 

(3) 

(4) 

c,= 

42 

60 

5w 

— 

c,= 

— 

50 

3n 

40 

R,= 

12 

— 

21 

8n 

R,= 

9 

15 

— 

6n 

12.  A  diamond  worth  $2000  was  broken  into  two  parts,  to- 
gether worth  only  $1600.  If  the  value  of  a  diamond  is  propor- 
tional to  the  square  of  its  weight,  into  what  fractions  was  the 
original  diamond  broken  ?     (Find  result  to  nearest  hundredth.) 


458  A   HIGH   SCHOOL  ALGEBRA 

LIMITS 

604.  In  the  construction  of  graphs  we  have  studied  the 
changes  in  functions  corresponding  to  given  values  of  their 
variables,  but  we  shall  now  consider  an  important  particular 
case  in  which  the  successive  values  of  a  variable  approach 
nearer  and  nearer  to  a  fixed  number. 

For  example,  if  we  take  the  values  of  the  decimal  .444  .  .  .  correspond- 
ing to  the  successive  decimal  places,  we  have  .4,  .44,  .444,  .4444,  .  .  .  ,  a 
series  of  increasing  numbers  so  limited  that  no  one  of  them,  however  far 
we  go,  can  be  as  large  as  .45.  In  fact,  no  one  of  them  can  equal  |,  but  by 
using  more  and  more  decimal  places  we  may  come  to  terms  differing  as 
little  from  f  as  we  choose. 

605.  Limit.  If  a  variable  x  has  a  boundless  number  of  suc- 
cessive values  which  approach  nearer  and  nearer  to  a  fixed 
number  I,  so  that  the  difference  I  —  x  may  become  and  remain 
numerically  as  small  as  we  choose,  x  is  said  to  approach  I  as  a 
limit. 

606.  This  relation  is  expressed  in  symbols  by  a?  =  Z,  read, 
"  X  approaches  Z  as  a  limit." 

For  example  :  In  Sec.  604,  if  the  variable  x  be  taken  to  represent  the 
different  numbers  in  the  series  .4,  .44,  .444,  .  .  .  ,  then  x  =  ^. 

Or,  if  X  is  successively  .02,  .002,  .0002,  and  so  on  as  far  as  we  choose, 
then  a;  =  0. 

607.  Meaning  of  ^.     We  have  shown  (Sec.  487,  II)  how  the 

indicated  operation,  -,  may  arise  from  solving  systems  of  equa- 
tions, and  that  its  value  is  indeterminate.  We  will  further 
illustrate  it  here  by  use  of  expressions  having  limiting  values. 

X—  1  0 

For  example,  the  fraction  — — -  becomes  -,    it  x  =  1,  but  it  can  be 

shown  to  have  the  limiting  value  |.    If  we  let  x  =  .9,  .99,    .999,   then 

x-l  ^  10     100     1000  .....  u  1         ^ 

becomes  — r,    -— -,    z-r-:,   a  series  of  fractions  whose  values  de- 


jc2_i  '    19'    199'    1999' 

crease  toward  I  when  x  =  1.  Furthermore,   -^  ~     = by  reducing  to 

x^  -1      X  +  1 

1-1  y 1 

lowest  terms,  and =  -,  when  x  =  1.     Thus,  has  the  limit- 

X  +  1       2  x^  —  1 

ing  value  I  when  a:  =  1. 


PROPORTION,  VARIATION,   AND  LIMITS  459 


Similarly,  the  fraction  — has  the  limiting  value  —  1,  when  x  —  0. 

Thus  the  expression  -  may  represent  different  numbers,  and  as  a 
symbol  taken  by  itself,  it  must  be  regarded  as  indeterminate. 

608.  Meaning  of  (».  Opposed  to  quantities  which  tend  to- 
ward zero,  are  quantities  which  grow  large  without  bound. 
Such  variables  are  said  to  have  the  property  of  becoming 
infinitely  great. 

For  example,  if  x  assumes  in  turn  all  integral  values,  1,  2,  3,  4,  and  so 
on,  without  end,  x  is  said  to  have  the  property  of  becoming  infinite. 

The  symbol,  oo,  called  infinity ,  is  commonly  used  to  express 
the  fact  that  a  variable  has  the  above  property,  but  it  must  not 
be  regarded  as  a  particular  number  nor  as  a  limit;  it  is  merely 
a  sign  of  the  infinitely  great. 

Thus,  n  =  CO  means  "  when  u  becomes.infinitely  great." 

609.  Meaning  of  if..     This  means  a  fixed  number  divided  by 

00 

a  variable  number  which  grows  large  without  bound.     Under 
these   conditions,    ^^^O. 

CO 

For  example,  let  a  be  represented  by  a  line  1  ft.  long,  and  suppose  it  to 
be  bisected.     The  result  is  '-  =  6  in.      Then,  suppose  each  division 

to  be  bisected  again.     The  result  is —  =  3   in.      Bisect  each  division 

4 
again,  and  suppose  the  process  to  be  continued  indefinitely.     The  denom- 
inator becomes  32,768  when  the  division  has  been  made  15  times,  and 

— ^is  less  than in.     By  taking  more  and  more  divisions,  the  frac- 

32,768  2730  ^  ^ 

tion  expressing  the  length  of  a  division  approaches  zero  as  a  limit. 

610.  Meaning  of  ^.  This  means  the  quotient  of  two  vari- 
able numbers  each  of  which  grows  large  without  bound,  and 
the  expression  can  have  no  fixed  value.  But  the  quotient  may 
tend  to  a  limit  as  both  numerator  and  denominator  increase. 

For  example,  when  x  =  cc,  — - —  =  ^,  but  it  has  the  limiting  value  1, 
1  +x      °°  11 

as  may  be  seen  by  dividing  both  terms  by  x  ;  thus,  — - —  = = 

1  +  x      ^_^1      1+0 

or  1,  when  x  =  ixi.  x 


460  A  HIGH   SCHOOL   ALGEBRA 

WRITTEN    EXERCISES 

What  is  the  limiting  value  of  each  of  the  following  when 
w  =  oo : 

1.  i.  3.    5.  6.    '-+1 
X                                y  z 

2.  '5.  4.    -y—.  6.     ^-^. 
X                                2/  — 2  2 

a;  (a;  4-  2)     (Eeduce  and  separate  into  a  whole  number 
a^  and  a  fraction  before  substituting  a;  =  oo.) 

8.  Apply  the  suggestion  in  Ex.  7  to  a?(a;- l)(a;-2)^ 
What  value  does  each  expression  approach  as  v  ^  0 : 

9.  Ip  10.    2*  11.    3^29  12.    _JL_? 
--  v  +  l 


V 


Pind  the  limiting  value  of :  " 

13.  ^as.  =  2.  15.    ^  +  ^^-^as^=^L 
^2-4  ar'-l 

14.  ?^^  as  2  =  3.  16.        ^~^     asa;=-2. 
2-3  a;2  4-aj-2 


GRAPHICAL  WORK 

611.  The  relation  "a;  varies  as  y  "  has  been  expressed  by  the 
linear  equation  x  =  ky,  and  is  represented  by  a  straight  line 
(Sec.  287,  p.  205). 

The  relation  "a;  varies  inversely  as  2/"  has  been  expressed 

by  the  equation  x  =  -.     The  graph  of  this  equation  is  a  curve. 

y 


PROPORTION,   VARIATION,   AND   LIMITS  461 

2 
Suppose,  for  illustration,  that  k  =  2.    Then  x  =  -.     For  this  equation : 


The  table  is 


5 
4 
3 
2 
1 

\ 
-1 
-2 
-3 
-4 
-6 


1 

2 
4 

-2 
-1 

-f 
-1 


2/ 
The  graph  is 

y 


4 
3 

. 

- 

2 

'       1 

V     :    ,    ;  • 

i 

^^-— ^ 

■X 

^      -4     -3     -2      -1        0 

12        3        4 

1     ix  "^ 

1    nT' 

= 

ititf 

• 

1 1 1 ' 

/' 

If  large  numerical  values  are  given  to  y,  x  will  become  correspondingly 
small  so  that  for  y  =  cx),  a;  =  0.  Thus,  the  curves  approach  nearer  and 
nearer  to  the  y-a,xis.  Similarly,  when  x  =  oo,  then  y  =  0,  and  the  curves 
approach  nearer  and  nearer  to  the  x-axis. 


WRITTEN     EXERCISES 

1.    Under  standard  conditions  the  volume  (y)  of  a  confined 

k 
gas  varies  inversely  as  the  pressure  (p).     That  is,  v  =  -.    This 

is  known  as  Boyle's  Law.     Suppose  when  the  piston  is  at  a  in 

the  figure,  the  volume  of  the  B 

gas   in   part  ^  is  1   cu.    ft., 

and   the    pressure    at    P    is 

5  lb.     When  the  pressure  is 

10  lb.  the  piston  moves  to  b, 

and  the  volume  of  the  gas  B 


becomes  ^  cu.  ft. 
P=2aib.? 


What  will  be  the  volume  of  the  gas  when 


462 


A  HIGH  SCHOOL  ALGEBRA 


Taking  k  =  5m  the  equation  v  =  -,  the  table  of  values  is 

P 


p  = 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

15 

20 

v  = 

5 

21- 

If 

n 

1 

f 

f 

f 

f 

i 

i 

i 

The  graph  of  the  table  i= 


'■    * 

y 

. 

•   " 

5 
4 
3 

\ 

\ 

- 

1 |- 

2 

1 

v_______ 

0 

~^ 

1       2 

"T- 

4       5 

6 

7 

8-9 

10    1 

■'12' 

TT 

14   '15     18     17 

-18    19     20    ^ 

Eead  from  the  graph  the  pressure  on  the  piston  necessary  to  hold  this 
volume  of  gas  at  |  cu.  ft.  ;  at  |  cu  ft.  ;  at  5  cu.  ft. ;  at  2  cu.  ft. 
What  is  the  limiting  value  of  v  when  p  =  cc? 


Eepresent  graphically : 

2.  x  =  -,  or  xy  =  1. 

y 

2 

3.  x  = ,  oric?/  =  — 2. 

y 


4.  x  =  i 

y 

5.  3  07  = 


,ovxy  —  4. 
-1 


,  ov  2»  xy^  —  1. 


6.  The  attraction  or  "  pull "  of  the  earth  on  bodies  in  its 
neighborhood  is  the  cause  of  their  weight.  The  law  of  gravita- 
tion states  that  the  weight  of  a  given  body  varies  inversely  as 
the  square  of  its  distance  from  the  center  of  the  earth.     This 

k 
law  may  be  expressed  by  the  equation  z^  =  —  • 

To  construct  the  graph  which  shows  the  nature  of  the  relation  between 
w  and  d  as  they  vary,  yfc  may  be  taken  to  be  1. 

Fill  the  blanks  in  the  table : 


d 

±  1 

±5 

±\ 

±  .25 

-J:.2 

w 

1 

(     ) 

(     ) 

(      ) 

(      )* 

7.   Plot  the  graph  for  the  table  in  Exercise  6. 


PROPORTION,  VARIATION,   AND   LIMITS  463 

REVIEW 

WRITTEN   EXERCISES 

1.    Solve,  using  the  principle  of  composition  and  division, 


{a-^2,hx  +  a?)  :  (a  -  6)  =  (a  +  V2  6a; -f  a;^)  :  (a  +  6). 

2.  If    ^i  =  ^2^«2^...^«,^pj,o^etliat 

6l  62  ^3  ^n 

&l  +  62  +  &3+--+^«  ^l" 

3.  Given    a;  + Va; :  a;  — Va;::  3Va;  + 6  :  2Va;,  to  find  ic. 

4.  If  the  volume  of  a  sphere  varies  as  the  cube  of  its  radius, 
find  the  radius  of  a  sphere  whose  volume  equals  that  of  the  sum 
of  two  spheres  whose  radii  are,  respectively,  6  ft.  and  3.5  ft. 

5.  The  number  of  vibrations  (swings)  made  by  two  pen- 
dulums in  the  same  time  are  to  each  other  inversely  as  the 
square  roots  of  their  lengths.  If  a  pendulum  of  length  39  in. 
makes  1  vibration  per  second  (called  a  seconds  pendulum), 
about  how  many  vibrations  will  a  pendulum  10  in.  long  make  ? 
How  long  must  the  pendulum  be  to  make  10  vibrations  per 
second  ? 

6.  Two  towns  join  in  building  a  bridge  which  both  will  use, 
and  agree  to  share  its  cost,  $5000,  in  direct  proportion  to  their 
populations  and  in  inverse  proportion  to  their  distances  from 
the  bridge.  One  town  has  a  population  of  5000  and  is  2  mi. 
from  the  bridge;  the  other  has  a  population  of  9000  and  is 
6  mi.  from  the  bridge.     What  must  each  pay  ? 

3  .  .    .  . 

7.  In  a;  =  -  what  is  the  limiting  value  of  x  when  y  =  cc? 

y 

8.  In  a;  =  J  what  does  x  become  when  y  =  GO? 


rf^  25 

9.   Find  the  limiting  value  of  as  a;  =  —  5. 

^  x''  +  2x-15 


464  A   HIGH   SCHOOL  ALGEBRA 

SUMMARY 

The  following  questions  summarize  the  definitions  and  pro- 
cesses treated  in  this  chapter : 

1.  Define  Si proportion ;  define  means;  also  extremes. 

Sees.  584,  588. 

2.  What  is  a  fourth  proportional  ?     A  third  proportional  ? 
A  mean  proportioyial  9  Sees.  585-587. 

3.  Define  alternation ;  also  inversion ;   composition ;   divi- 
sion. Sees.  590-594. 

4.  What  is  a  continued  proportion  ?  Sec.  595. 

5.  Define  and  illustrate  direct  variation.  Sec.  597. 

6.  State  the  relation  of  variation  to  proportion.      Sec.  599. 

7.  Define  and  illustrate  inverse  variation.  Sec.  601. 

8.  State  an  equation  expressing  the  law  of  direct  variation  ; 
also  one  expressing  inverse  variation.  Sees.  600,  602. 

9.  Define  and  illustrate  limit.  Sec.  605. 

10.  Explain  the  meaning  of  the  symbol  - ;  also  oo ;  also 
^;  also  -.  .  Sees.  607-610. 

00  00 

11.  What  kind  of  a  line  is  the  graph  of  the  equation  express- 
ing direct  variation?  Of  the  equation  expressing  inverse 
variation?  Sec.  611. 


CHAPTER   XXXII 

SERIES 

612.  Series.  If  a  sequence  of  numbers  is  determined  by  a 
given  law,  the  sequence  of  numbers  is  called  a  series. 

613.  Terms.  The  numbers  constituting  the  series  are  called, 
its  terms,  and  are  named  from  the  left,  1st  term,  2d  term,  etc. 

The  following  are  examples  of  series : 

1.    1,2,3,4,5,....  7.   hhhhh-" 

8.  1,  3,  9,  27,  81,  243,  .... 

Q       2     2      2.       2      ... 
10-     1?    2?  "g"?  T'  Zf  7>  '**• 

11.  V2,  V3,  Vi,  V5,  V6,  Vf,  .... 

12.  100,99,98,  97,96,95,  .... 

ORAL    EXERCISES 
1-12.     State  the  next  five  terms  of  each  series  above. 

614.  When  the  law  of  a  series  is  known,  any  term  may  be 
found  directly. 

EXAMPLES 

1.  The  law  of  the  second  series  in  Sec.  613  is  that  each  term  is  two 
more  than  the  preceding.  To  get  the  tenth  term  we  start  from  1  and 
add  2  for  each  of  9  terms. 

That  is,  the  tenth  term  is  1  +  9  •  2  =  19. 

Similarly,  the  12th  term  is  1  +  11  •  2  =  23, 
■  the  15th  term  is  1  +  14  .  2  =  29, 
the  47th  term  is  1  4-  46  •  2  =  93, 
the  nth  term  is  1  +  (w  —  1)2  =  2  w  —  1. 
465 


2. 

1,3,5,7,9,... 

3. 

1,  5,  9,  13, 17, 

4. 

3,  6,  9,  12,  15, 

5. 

1,  li,  2,  21  3, 

6. 

2,  4,  8,  16,  32, 

466         A  HIGH  SCHOOL  ALGEBRA 

[2.  The  law  of  the  eighth  series  is  that  each  term  after  the  first  is  3 
times  the  preceding  term.  To  get  the  ninth  term  we  start  at  1  and 
multiply  by  3  eight  times,  or  by  S^.    That  is,  the  ninth  term  is  1  •  3^  =  6561. 

Similarly,  the  12th  term  is  1  •  311  =  177,147, 
the  nth  term  is  1  .  3"-i  =  3"-!. 

WRITTEN     EXERCISES 

1-4.  Select  four  of  the  series  in  Sec.  613  that  can  be  treated 
like  the  first  example  and  write  the  10th,  12th,  15th,  and  47th 
terms  of  each. 

5-7.  Select  three  of  the  series  in  Sec.  613  that  can  be  treated 
like  the  second  example  and  write  the  8th,  10th,  and  nth  terms 
of  each. 

8.  Write  similarly  the  7th,  11th,  20th,  47th,  and  nth.  term 
for  any  6  of  the  above  series. 

615.  We  shall  give  only  two  types  of  series,  the  arithmetical 
and  the  geometric,  the  laws  of  which  are  comparatively  simple. 

ARITHMETICAL  SERIES 

616.  Arithmetical  Series.  A  series  in  which  each  term  after 
the  first  is  formed  by  adding  a  fixed  number  to  the  preceding 
term  is  called  an  arithmetical  series  or  arithmetical  progression. 

617.  Common  Difference.  The  fixed  number  is  called  the 
common  difference,  and  may,  of  course,  be  negative. 

For  example  : 

1.  7,  15,  23,  31,  39,  •••  is  an  arithmetical  series  having  the  common  dif- 
ference 8, 

2.  16,  141,  13,  11|,  10,  •••  is  an  arithmetical  series  having  the  common 
difference  —  f . 

WRITTEN     EXERCISES 

1.  Select  the  arithmetical  series  in  the  list  of  Sec.  613. 
Write  the  nth  term  in  each. 

2.  Beginning  with  2  find  the  100th  even  number. 

3.  Beginning  with  1  find  the  100th  odd  number. 


SERIES  467 

4.  Beginning  with  3  find  the  200th  multiple  of  3. 

5.  A  city  with  a  population  of  15,000  increased  600  persons 
per  year  for  10  yr.  What  was  the  population  at  the  end  of 
10  yr.? 

618.  A  general  form  for  an  arithmetical  series  is : 

a,  a-\-  d,  a-\-2  d,  a-\-S  d,  •  •  •,  a  +  (?i  —  l)d,  •  •  •, 
where  a  denotes  the  first  term, 

d  denotes  the  common  difference,  and 
n  denotes  the  number  of  the  term. 

619.  Last  Term.  If  the  last  term  considered  is  numbered  n 
and  denoted  by  I,  we  have  for  the  last  of  w  terms  the  formula: 

I  =  a -\- (71 —l)d. 

620.  The  Sum  of  an  Arithmetical  Series.     The  sum  of  n 

terms  of  an  arithmetical  series  can  be  found  readily. 

EXAMPLE 

Find  the  sum  of  the  first  6  even  numbers. 

1.  Let  s  =  2  +  4  +  6  +  8  +  10  +  12. 

2.  We  may  also  write  s  =  12  +  10  +  8  +  6  +  4  +  2. 

3.  Adding  (1)  and  (2), 

2s=(2  +  12)  +  (4  +  10)  +  (6  +  8)  +  (8  +  6)  +  (10+4)  +  (12+2) 
=  6(2  +  12) ;  for  each  parenthesis  is  the  same  as  2  +  12. 

4.  .,,^6(2  +  12)^^^^ 

2 

WRITTEN    EXERCISES 

Eind  similarly  the  sum  of : 

1.  The  first  6  odd  numbers. 

2.  The  first  6  multiples  of  3. 

3.  The  first  5  multiples  of  7. 

4.  The  first  4  multiples  of  8. 

621.  General  Formula  for  the  Sum.  The  general  form  of 
the  series  may  be  treated  in  the  same  way. 


468  A  HIGH  SCHOOL  ALGEBRA 

If  I  denotes  the  last  of  n  terms,  the  term  before  it  is  denoted 
hj  l  —  d,  the  next  preceding  hyl  —  2d,  and  so  on.  Hence,  the 
sum  of  n  terms  may  be  written : 

s  =  a  +  (aH-d)  +  (a+2d)H y-{l-2  d)  +  {l-d)^l. 

And  also,  s=^l+{l—d)  +  {l-2d)-\ \-{a^2  d)  +  {a+d)-[-a. 

Whence,  adding,  2  s={a-\-l)-\-{a  +  l)-\ \-{a+l)  =  n{a-\-l). 

Therefore,  g  =  ^^^  +  ^). 

Or,  in  words. 

The  sum  of  any  number  of  terms  of  an  arithmetical  series  is  one 
half  the  sum  of  the  first  and  the  last  terms  times  the  number  of 
terms. 

By  using  the  value  of  I  in  Sec.  619,  s  =  ^-^ — ^      ^  ~  * 

This  permits  the  calculation  of  s  without  working  out  separately  the 
value  of  I. 

WRITTEN     EXERCISES 

For  each  series  in  the  following  list  find :  First,  the  sum  of 
10  terms.     Second,  the  sum  of  n  terms. 

1.  1,  2,  3,  4,  5,  ....  4.   3,  6,  9,  12,  15,  .... 

2.  1,3,5,7,9,....  5.   1,  li,2,2i,3,  .... 

3.  1,  5,  9,  13,  17,  ....         6.    100,  99,  98,  97,  96,  95,  .... 

7.  A  man  invests  $100  of  his  earnings  at  the  beginning  of 
each  year  for  10  yr.  at  6  %,  simple  interest.  How  much  has 
he  at  the  end  of  10  yr.  ? 

Solution. 

1.    The  last  investment  bears  interest  1  yr.  and  amounts  to  %  106  ;  the 
next  to  the  last  bears  interest  2  yr.  and  amounts  to  $112,  etc.;  the  first 
bears  interest  10  yr.  and  amounts  to  $  160. 
•     2.    Hence,  a  =  $  106,  d  =  %^,  and  n  =  10. 

3.  Therefore,  Z  =  106  +  9  •  6  =  160. 

4.  Therefore,  s  =  ^Q.  (106  +  160)  =  1330. 

5.  The  man  has  $1330. 

8.  If  $  50  is  invested  at  the  beginning  of  each  year  for  20  yr. 
at  5  %  simple  interest,  what  is  the  amount  at  the  end  of  20  yr.  ? 


SERIES  469 

9.  If  a  body  falls  approximately  16  ft.  the  first  second  and 
32  ft.  farther  in  each  succeeding  second,  how  far  does  it  fall 
in  5  sec.  ? 

622.  Collected  Results.  The  three  chief  formulas  of  arith- 
metical series  are : 

1.    l  =  a-^(n-l)d. 

2 
3    g  __  n  [2  g  +  (^  —  1)  dl 
2 

GEOMETRIC   SERIES 

623.  Geometric  Series.  A  series  in  which  each  term  after 
the  first  is  formed  by  multiplying  the  preceding  term  by  a  fixed 
number  is  called  a  geometric  series,  or  a  geometric  progression. 

624.  Common  Ratio.  The  fixed  multiplier  is  called  the  com- 
mon ratio,  and  may  be  negative. 

For  example : 

1.  2  is  the  common  ratio  in  the  geometric  series  2,  4,  8,  16,  •••. 

2.  —  ^  is  the  common  ratio  in  the  geometric  series  27,  —  9,  3,  —  1,  |,  •••. 

ORAL    EXERCISES 

1.  Name  all  the  geometric  series  in  the  list,  Sec.  613. 

2.  State  the  common  ratio  in  each  of  these  series. 

3.  State  the  6th  term  of  each  of  these  series. 

WRITTEN    EXERCISES 

1.  The  series  1,  3,  9,  27,  81,  •••,  whose  ratio  is  3,  is  the  same 
as  1,  3^,  3^,  3^,  3'',  •••.  Write  by  use  of  exponents  the  6th  term 
of  this  series ;  the  8th  term ;  the  10th ;  the  15th ;  the  100th. 

2.  The  series  3,  —  f ,  f,  —  f,  •••,  whose  ratio  is  —  i,  is  the 
same  as  3,  3(-i),  3(-i)2,  3(-i)^  ....  Write  by  use  of  ex- 
ponents the  5th  term  of  this  series ;  the  8th  term ;  the  10th ; 
the  25th ;  the  50th. 


470  A   HIGH   SCHOOL   ALGEBRA 

625.  A  general  form  for  the  geometric  series  is : 

a,  ar,  ar^,  ai^,  ••-,  ar""^  •••, 

where  a  denotes  the  first  term, 

r  denotes  the  common  ratio,  and 
n  denotes  the  number  of  the  terras. 

626.  Last  Term.     If  the  last  term  is  numbered  n,  and  de- 
noted by  I,  then  we  have  for  the  last  of  n  terms  the  formula, 

627.  The  Sum  of  a  Geometric  Series.     The  sum  of  n  terms 
of  a  geometric  series  can  readily  be  found. 

EXAMPLE 

Find  the  sum  of  5  terms  of  the  series  2,  6,  18,  54,  162. 
Solution. 

Let  s  ^  2  +  6  +  18  +  54  +  162.  (i) 

""^iSSo!' '''  3  .  =  6  +  18  +  54  +  162  +  486.  (^) 

Subtracting  (I)  from  (;?),      Ss  —  S  =  486  —  2,  (3) 

or,  2  s  =  484.  (4) 

Dividing  by  2,  S  =  242.  (5) 


WRITTEN    EXERCISES 

Find  similarly  the  sum  of  5  terms  of  each  of  these  series : 


1.   6,30,150,....  3.   hT\,to,'" 


1 

50 

2.    7,-14,28,....  4.   -i,  -i  tV.  •••• 


628.    General  Formula  for  the  Sum.     The  general  form  of  the 
series  may  be  treated  in  the  same  way.     If  I  denote  the  last 

of  n  terms,  the  term  before  it  is  denoted  by  -,  the  next  preceding 

r 

by  — ,  and  so  on.     Hence,  the  sum  of  7i  terms  may  be  written : 
1.    s  =  a  -f-  ar  +  «^'^  +  —  ^  H \-l- 


SERIES  471 

2.  Then  rs  =  ar  +  ar2  H L-^l^i^-ir. 

3.  Subtracting,      s  —  rs  =  a  —  Ir. 

4.  Or,  (1  —  r)  s  =  a  —  Ir. 

_a  —  Ir _  Ir  —  a ^ 
1  —  r       r  —  1 
In  words, 

The  Slim  of  any  number  of  terms  of  a  geometric  series  is  the 
ratio  times  the  last  term  diminished  by  the  first  term  and  divided 
by  the  ratio  less  1. 

By  using  the  value  of  I  (Sec.  626), 

_  ar"~^  •  r  —  g  _  qr"  —  g    ^ C^  / /) 


'  ^-./) 


^-/ 


r  —  1  r  —  1 

Thus,  s  may  be  found  without  first  computing  I. 

WRITTEN    EXERCISES 
1    Find  the  sum  of  6  terms  of  the  series  |,  ^,  J,  ••... 
2.    Find  the  sum  of  10  terms  of  the  series  ^,  —  J,  |,  •••. 
3    Find  the  sum  of  8  terms  of  the  series  1,  .25,  .0625,  •••. 

4.  Find  the  sum  of  12  terms  of  the  series  27,  —  9,  3,  —  1,  •••. 

5.  An  air  pump  exhausted  the  air  from  a  cylinder  containing 
1  cu.  ft.  at  the  rate  of  J^  of  the  remaining  contents  per  stroke. 
What  part  of  a  cubic  foot  of  air  remained  in  the  cylinder  after 
25  strokes  ? 

6.  The  population  of  a  town  increased  from  10,000  to 
14,641  in  5  yr.  If  the  population  by  years  was  in  geometric 
series,  what  was  the  rate  of  increase  per  year  ? 


629    Collected  Results. 

The  three  chief  formulas  of  geo- 

metric  series  are : 

1. 

l  =  ar--\ 

2. 

s-'^--- 

3.   s  = 
31 


r-1 
«?'**  —  a 


472 


A   HIGH   SCHOOL   ALGEBRA 


1st 

$  100  (1.05) 

2d 

$  100  (1.05)'^ 

3d 

$100  (1.05)3 

4th 

$100  (1.05)* 

WRITTEN     EXERCISES 

1.  $100  is  placed  on  interest  at  5  %,  compounded  annually. 

(1)  What  is  the  amount  at  the  end  of  the  first  year  ? 

(2)  What  is  the  principal  for  the  second  year  ? 

(3)  What  is  the  amount  at  the  end  of  the  second  year  ? 
Notice  that  the  amounts  appear  in 

the  right-hand  column  of  the  table. 

(4)  Indicate  similarly  the  amount 
of  $  100  at  the  end  of  5  yr. ;  10  yr. ;  n 
yr.  Which  formula  of  geometric 
series  expresses  the  amount  for  ri  yr.  ? 

2.  Indicate  the  amount  of  $100  at  6%,  compounded  an- 
nually, at  the  end  of  1  yr. ;  2  yr.  ;  5  yr. ;  10  yr. ;  n  jv . 

3.  Many  savings  banks  pay  interest  at  the  rate  of  3%, 
compounded  semi-annually. 

Indicate  the  amount  of  $  100  under  the  above  conditions  at 
the  end  of  6  mo. ;  1  yr. ;  18  mo. ;  2  yr. ;  10  yr. ;  n  yr. 

Note.  The  numerical  value  of  these  expressions  can  be  computed 
readily  by  logarithms. 

Solve  by  the  use  of  logarithms  : 

4.  What  is  the  amount  of  $  1  at  4  %  compound  interest  for 
8  yr.  ? 

5.  A  man  deposits  $  100  in  a  bank  paying  4  %  interest,  com- 
pounded annually,  on  the  first  day  of  each  year  for  5  yr. 
How  much  will  he  have  on  deposit  at  the  end  of  5  yr.  ? 

6.  Determine  similarly  the  amounts  of : 


Deposit  at  Beginning 

Rate  op  Interest  Com- 

Number of  Years 

OF  Each  Year 

pounded  Annually 

(1) 

,  $25 

5 

8 

(2) 

10 

6 

15 

(3) 

43.20 

4 

20 

(4) 

39.87 

3 

20 

SERIES  473 


MEANS 


630.  Means.  Terms  standing  between  two  given  terms  of 
a  series  are  called  means. 

631.  Arithmetical  Mean.  If  three  numbers  are  in  arith- 
metical progression,  the  middle  one  is  called  the  arithmetical 
mean  between  the  other  two. 

The  arithmetical  mean  between  a  and  h  is  found  thus : 

1.  Let  A  be  the  mean  and  d  the  common  difference. 
Then  the  terms  may  be  written  A  —  d  and  A  +  d. 

2.  Whence,        A  —  d=ia  and  A  +  d  =  h. 

3.  Adding,  2  ^  =  a  +  &  and  ^  =  ^-±-^  • 

632.  The  arithmetical  mean  between  two  numbers  is  one  half 
their  sum. 

633.  Geometric  Mean.  If  three  numbers  are  in  geometric 
progression,  the  middle  one  is  called  the  geometric  mean  be- 
tween the  other  two. 

The  geometric  mean  between  a  and  b  is  found  thus : 

1.  Let  g  be  the  geometric  mean. 

2.  Then  ^  =  ^. 

a     g  _ 

3.  .  •.  g^  =  ah  and  g  =  y/ah. 

634.  The  geometric  mean  between  two  numbers  is  the  square 
root  of  their  product.  There  are  really  ttvo  geometric  means,  one 
negative  and  one  positive. 

The  geometric  mean  between  two  numbers  is  the  same  as 
their  mean  proportional. 

ORAL     EXERCISES 

State  the  arithmetical  mean  between : 

1.   8,  12.  2.   6,  3.  3.   4,  -  10.  4.   5  a,  13  a. 

State  the  geometric  mean,  including  signs,  between : 

5.    8,  6.  6.   3,  12.  7.    a,  a\         •        S.   2y?,  32  x'. 


474  A  HIGH  SCHOOL  ALGEBRA 

635.  Any  number  of  means  may  be  found  by  use  of 
formulas  already  given. 

EXAMPLES 

1.  Insert  5  arithmetical  means  between  4  and  12. 

1.  In  this  case  a  =  4,  I  =  12,  and  n  =  l. 

2.  .'.  I  =  a  +  {n  —  l)d  becomes  12  =  4  +  (7  —  l)d. 

3.  Solving  for  d,  d  =  — ^^-  =  l\. 

4.  Adding  1^  to  4,  and  1^  to  that  result,  and  so  on,  the  means 

are  found  to  be  5i,  6|,  8,  9i,  and  lOf . 

2.  Insert  4  geometric  means  between  —  27  and  ^. 

1.  In  this  case  a  =  —  27,  1  =  1,  and  n  =  6. 

2.  .-.  I  =  ar'^-i  becomes  I  =  —21  r*. 

3.  Solving  for  r,r^  =  -  ^-^  =  ~  p  * 

Therefore,  r  =— ^. 

4.  Multiplying  —  27  by  —  -i,  and  multiplying  this  result  by  —  i, 

and  so  on,  the  means  are  found  to  be  9,  —  3,  1,  and  —  ^. 

WRITTEN     EXERCISES 

1.  Insert  3  arithmetical  means  between  6  and  26. 

2.  Insert  10  arithmetical  means  between  —  7  and  144. 

3.  Insert  3  geometric  means  between  2  and  32. 

4.  Insert  4  geometric  means  between  —  -^q  and  3^. 

OTHER  FORMULAS 

636.  Arithmetical  Series.  By  means  of  the  formulas  of 
Sec.  622,  any  two  of  the  five  numbers  a,  n,  I,  d,  s,  can  be  found 
when  the  other  three  are  given. 

EXAMPLES 

1.    Given  n  =  6,  s  =  18,  Z  =  8,  find  a,  d. 

1.   For  these  values,  formulas  (1)  and  (2)  become  : 
8  =  a+(6-l)d, 
18  =  ^(«  +  8) 


SERIES  475 

2.  We  have  thus  two  equations  to  determine  and  two  num- 

bers, a,  d. 
From  the  second  equation,  a  =  —  2. 

3.  Using  this  value  in  the  first  equation,  d  =  2. 

2.  Given  a  =  4,  1  =  12,  s  =  56,  find  n  and  d. 

1.  For  these  values  formulas  (1)  and  (2)  become  : 

12  =  4  +  (w  -  l)d 

56=^^^  +  ^^). 

2 

2.  .-.  from  the  second  equation,  w  =  7. 

3.  Substituting  in  the  first,  12  =  4  +  6  d, 

4.  therefore  (?  =  f . 

3.  Given  w  =  12,  s  =  30,  Z  =  10,  find  a,  d. 

1.  Formulas  (1)  and  (3)  become  : 

10  =  a  +  nd. 

2 

2.  Solving  these  equations  for  a  and  d : 

a  =r  —  5,  and  <?  =  ^. 

The  same  results  would  be  found  by  using  formulas  (1)  and  (2),  since 
(3)  is  only  another  form  of  (2). 

637.   The  same  problems  can  also  be  solved  generally ;  that 
is,  without  specifying  numerical  values. 

EXAMPLE 

Eegarding  n,  s,  d  as  known,  find  a,  I. 

1.  From  (1),  Sec.  622,       I-  a=(n-  l)d. 

2.  From  (2),  Sec.  622,       a  +  l=^- 

n 

3.  Adding  (1)  and  (2),         2Z=(n-l)d+— • 

n 

Or,  i  =  (JLz^)^^t. 

2  n 

4.  Substituting  in  (2)  above,  a=  —  -  ['IZLizlM  + 11 

?i       L       2  nj 

_s      (,n--\)d 


476 


A  HIGH   SCHOOL  ALGEBRA 


WRITTEN   EXERCISES 

By  use  of  the  formulas  in  Sec.  622,  find  the  following 


5. 
6. 

7. 

8. 

9. 

10. 

IL 
12. 

13. 
14. 
15. 
16. 

17. 
18. 
19. 
20. 


Find  In  Terms  of 


adn 
ads 
ans 

dns 


adn 
a  d  I 

a  n  I 
dn  I 


d  n  I 
dns 

dls 
n  Is 


anl 
ans 
als 
n  Is 


adl 
ads 
als 
d  Is 


Eesult 


I  =  a  -\-  (^n  —  l)d 


l  =  ^[-d±V8ds-\-(_2a-  d)2] 

;       2  s 

I  = a 

n 

l^s__^  {n-\)d 

n  2 


s  =  \  n{2  a  +  {n  -  I)  d'] 

s  =  ^  +  0^  +  ^^  -  «' 


2d 


=  ~(a-\-  I) 


s  =  ln[2l-(n-l)d] 


a  =  1  —  (n—  l)d 
^^s      (n-l)d 


a  =  l[d±  V(2  I  +  d)-^  -  8  ds] 

2s      , 
a  =  —  —  I 


d  =  l^^ 

71-1 

^  ^  2  (s  -  an) 
w  (w  —  1 ) 
I'^-aP- 


d  = 


2s-l  -a 

2  (nl  -  s) 


n  (n  -  1) 

d 

d~2a±V(2a- 

-  rf)2  +  8  ds 

2d 


I  +  a 


^^^2l  +  d±  V(2  l  +  d)^-8d8 
2d 


Note,    a,  I,  d,  s  may  have  any  values,  but  n  must  be  a  positive  integer. 
Hence,  not  all  solutions  of  equations  18,  20,  represent  possible  series. 


SERIES  477 

638.  Geometric  Series.  By  means  of  the  formulas  of  Sec. 
629,  any  two  of  the  five  numbers  a,  n,  I,  r,  s,  can  be  found 
when  the  other  three  are  given. 

EXAMPLES 

1.  Given  s  =  1024,  r  =  2,  a  =  2,  find  I. 

1.  For  these  values  formula  (2)  becomes  : 

2-1         ^         ^ 

2.  .-.  I  =  513. 

2.  Given  r  =  5,  w  =  5,  s  =  363,  find  a. 

1.  For  these  values  formulas  (1)  and  (2)  become  : 

Z  =  a  .  34. 

363  =  ^  ^  ~  ^  • 
2 

2.  Eliminating  Z,       363^«(3^-l). 

3.  Therefore,  363  =^  a  •  242,  and  a  =  3. 

3.  Given  s  =  363,  a  =  3,  r  =  3,  find  n. 

1.  For  these  values  formula  (3)  becomes  : 

363=^  -3" -3^ 
2 

2.  Therefore,  3"  -  1  =  242,  and  3"  =  243. 

3.  By  factoring  243,  n  is  seen  to  be  5. 

Note.  In  finding  n  it  may  not  be  possible  to  factor  as  in  the  case  of 
243  above.    In  this  case  logarithms  may  be  applied. 

639.  The  same  problems  can  be  solved  generally,  that  is, 
without  specifying  numerical  values. 

EXAMPLE 

Express  I  in  terms  of  a,  n,  and  s. 

1.  From  formula  (2)  r  =  ^^^^,  or  r"-i  =  (^  -  q)^~\ 

^  ^         s-l  (s  -  0""^ 

2.  .-.  substituting  in  (1)  1  =  Q'C^-^)""^ . 

3.  .-.  Z(s_  ?)«-!_  a(s -«)"-!  =  0. 

This  equation  is  of  a  degree  higher  than  2  in  I  when  w  >  3.  But  for  n 
equal  to  or  less  than  3  it  can  be  solved  by  methods  already  explained. 


478 


A   HIGH   SCHOOL   ALGEBRA 


WRITTEN    EXERCISES 

By  use  of  the  formulas  in  Sec.  629,  find  the  results  given  in 
the  table : 

Note.     In  Exercises  3,  12,  and  16,  only  the  equation  connecting  tlie 
unknown  numbers  with  the  given  ones  can  be  found  : 


Find 


In  Tekms  of 


Result 


10. 

11. 

12. 


13. 
14. 
15. 
16. 


am 
ars 
ans 
rns 


am 
arl 

aln 

ml 


rn  I 

ms 

rl  s 
n  I  s 


a  n  I 
ans 
a  I  s 
n  I  s 


1  = 
1  = 
l(s 
1  = 


ar^~^ 

g  +  (r—  l)s 

r 
-l)n-i_a{s-ay-^  =  Q 
(r  —  1)  sr"-^ 
rn  -I 


a(r"  —  1) 

r-\ 
rl  —  a 
r-\ 

n—\/-i-        n— 1/ — 

Zr"  -  I 


^—1 


a  = 
a(s 


I 

(r-l)s 
r"  —  1 
W—  (r  —  l)s 
_  a)»-i  _  z  (s  -  l)n-i  =  0 


^a 


a  a 

s  —  a 


-I 


-I  s-l 


SERIES  479 


SPECIAL   SERIES 

640.   Binomial  Expansion.    The  Binomial  Expansion,  Chapter 
XX,  is  a  series  in  which  each  term  of  {a  +  hY  is  produced  from 

the  next  preceding  one  by  multiplying  by  -  and  inserting  in 

the  numerator  and  the  denominator  the  next  factor  in  each 
sequence. 

For  example,  the  third  term  in  the  binomial  expansion  of 

(a  +  by  is  ^^^^~-^)a"-25^  and  the  fourth  term  is 

n(n—l){n-2)  ^^_^^^ 
1.2.3 

To  form  the  fourth  term  from  the  third,  note  that : 

1.  a"-262  X  -  =  a'^-sfts. 

a 

2.  The  numerator,  w(ii  —  l)(w— 2),  of  the  coeflacient  of  the  fourth 
term  has  one  more  factor  in  the  sequence  of  factors,  which  begins  witi 
n  and  decreases  by  1  each  time. 

3.  The  denominator,  1.2.3,  has  one  more  factor  in  the  sequence  of 
factors  which  begins  with  1  and  increases  by  1  each  time. 

4.  In  the  expansion  of  (a  —  6)",  the  even  terms  are  negative. 

Thus,  from  any  given  terra  of  the  binomial  series,  all  the  subsequent 
terms  can  be  written. 


WRITTEN     EXERCISES 

1.  The  fifth  term  of  a  binomial  expansion  is 

1.2.3.4  ^' 

Write  the  sixth  term ;  also  the  seventh  term. 

2.  The  sixth  term  of  a  binomial  expansion  is 

n(n-l)(7i-2)(r^-3)(n-4)  ^^_,^, 
1.2.3.4.5 

Find  the  eighth  term. 

3.  By  Sec.  325,  1 .  2  may  be  written  [2,  called /ac^onaZ  tioo, 
1.2.3  may  be  written  |3,   called  factorial  three,  and  so  on. 


480  A   HIGH   SCHOOL   ALGEBRA 

Write  the  denominators  of  the  coefficients  in  Exercise  1  as 
factorials.     Also  in  Exercise  2. 

4.  Suppose  n  is  8  in  Exercise  1.  What  do  the  sixth  and 
seventh  terms  become?  Suppose  n  is  10  in  Exercise  2,  what 
does  the  eighth  term  become  ? 

641.  Finding  the  rth  Term.  Any  term  of  the  binomial  series 
may  be  written  at  once  by  observing  the  general  form  of  the 
terms,  as  explained  in  Sec.  640. 

Observe  that  : 

1.  The  last  factor  in  the  numerator  of  each  coefficient  is  n 
minus  a  number  two  less  than  the  number  of  the  term. 

E.g.,  in  the  third  term  the  last  factor  is  n—l\  in  the  4th  it  is  n  —  2  ; 
in  the  5th,  w  —  3,  and  so  on. 

2.  The  denominator  is  the  factorial  of  the  number  one  less  than 
the  number  of  the  term. 

E.g.,   [2  in  the  third,  [3^  in  the  4th,  and  so  on 

3.  The  exponent  of  the  first  term  of  the  binomial,  or  a  in 
(a  ±  b)°,  is  n  minus  a  number  one  less  than  the  yiumber  of  the 
term  ; 

E.g.,  w  —  1  in  the  2d  term,  ?i  —  2  in  the  3d,  and  so  on. 

4.  The  exponent  of  b  is  one  less  than  the  number  of  the  term  ; 
E.g.,  1  in  the  2d  term,  2  in  the  3d,  and  so  on. 

5.  The  signs  of  all  terms  are  positive  in  (a  +  b)°.  In  (a  —  b)° 
the  odd  terms  are  positive,  and  the  even  terms  negative. 

EXAMPLES 

1.   Find  the  9th  term  of  {x  —  y)". 

By  (1)  the  numerator  of  the  coefficient  is p(p  —  l)(p  —  2)  •••  (p  —  7). 

By  (2)  the  denominator  of  the  coefficient  is  [8. 

By  (3)  the  exponent  of  x  is  p  —  8. 

By  (4)  the  exponent  of  y  is  8. 

By  (5)  the  sign  is  +  • 

Therefore,  the  9th  term  of  (x  —  yy  is 

4- P(p-^)(p-2)  •••  (p-7)       8  8 
[_8  ^  * 


SERIES  481 

2.   Find  the  rth  term  of  (a;  —  2  a)". 

By  (1)  the  numerator  of  the  coefficient  is  w(w  —  1)  •••  (n  —  r  —  2). 

By  (2)  the  denominator  of  the  coefficient  is|  r  -  1. 

By  (3)  the  exponent  of  x  is  n  —  r  —  1. 

By  (4)  the  exponent  of  (2  a)  is  r  —  1. 

By  (5)  the  sign  is  i,  according  as  r  is  odd  or  even. 

Therefore,  the  rth  term  of  (x  —  2  a) «  is 

i  n(ii-l).»>(n^r  +  2)  ^^r+i(2  aV-i 

\r—l 


WRITTEN    EXERCISES 

Expand : 
1.    (x-ay.  2.    (a-\-xy.  3.    (2  a  -  xy. 

Write  without  expanding  the  series : 

4.  The  5th  term  of  (a  -  2  by. 

5.  The  7th  term  of  (a  -  xy^. 

6.  The  middle  term  of  (a  +  xy\ 

7.  The  two  middle  terms  of  (2  a?  —  y^)^, 

8.  The  10th  term  of  (a  +  3  a^yK 

9.  The  qth  term  in  (i  a;  —  2  yy. 

10.  The  rth  term  in  (x  —y)^. 

11.  The  (r  +  l)st  term  in  (a  +  hy. 

12.  The  (r  -  5)th  term  in  (a  -  h^^K 

642.  Finite  Series.  So  far  we  have  treated  only  series  with 
a  fixed  number  of  terms.  A  series  which  comes  to  an  end  is 
called  a  finite  series. 

643.  Infinite  Series.  A  series  whose  law  is  such  that  every 
term  has  a  term  following  it  is  called  an  infinite  series. 

For  example : 

2,  5,  8,  11,  •••,  239  as  here  written  ends  with  239.  But  the  law  of  the 
series  would  permit  additional  terms  to  be  specified.  In  the  above  ex- 
ample, the  next  following  terms  would  be  242,  245,  etc.  It  is  obvious 
that  however  many  terms  may  have  been  specified,  still  more  can  be 
made  by  adding  3.     The  series  is  thus  unending. 


482  A  HIGH  SCHOOL  ALGEBRA 

Similarly,  all  of  the  series  so  far  considered  might  have  been  con- 
tinued by  applying  their  corresponding  laws. 

The  term  '  •  infinite  ' '  comes  from  the  Latin  infinitus,  and  is  here  used 
with  the  meaning  unending. 

If  the  coefficients  of  the  binomial  expansion  be  regarded  as  a  series, 

1    n   C^~^)     ri(n-l)(n-2)     n(n  -  l)(n  -  2)(n- 3) 
'     '       2!      '  "  3!  '  4! 

they  furnish,  when  w  is  a  positive  integer,  instances  of  series  that  come  to 
an  end  according  to  the  law  of  the  series.  If  w  =  3,  the  series  has  4 
terms,  and  it  n  =  10,  the  series  has  11  terms  ;  for  the  positive  integer  w, 
\t  has  n  4- 1  terms.  This  is  true  because  the  factors  n,  n  —  1,  n  —  2,  and 
so  on,  will  finally  in  the  (n  +  2)d  term  contain  n  —  n  or  zero.  There- 
fore the  series  has  w  +  1  terms. 

But  if  n  is  a  negative  integer  or  a  fraction,  none  of  the  factors,  n, 
w  —  1,  w  —  2,  and  so  on,  becomes  zero,  and  the  series  can  always  be  ex- 
tended farther.  That  is,  if  n  is  a  negative  integer  or  any  fraction,  the 
series  is  unending  or  infinite. 

644.  Infinite  Geometric  Series.  The  subject  of  infinite  series 
is  of  great  importance,  but  is  too  difficult  to  be  taken  up  here. 
We  shall  mention  simply  a  few  properties  of  infinite  geometric 
series  lohose  ratio  is  numerically  less  than  1. 

The  following  are  examples  of  such  series  : 
1-   4,  2,  1,  1,  i,  .... 

9       Q      3        3  3         .. 

3.  .5,  .05,  .005,  .0005,  .... 

4.  1,  -i  ^,  -ixV-^V-. 

State  the  ratio  and  the  next  three  terms  of  each  series. 

I.  TJie  terms  become  numerically  smaller  and  smaller.  Each 
term  is  numerically  smaller  than  the  one  preceding  it,  for  it  is 
a  fractional  part  of  it. 

II.  The  terms  become  numerically  small  at  will.  That  is, 
however  small  a  number  may  be  selected,  there  are  terms  in 
the  series  smaller  than  it,  and  when  r  is  numerically  less  than 
1,  the  term  aV"'^  may  be  made  numerically  small  at  will,  by 
taking  n  sufficiently  large. 

This  seems  obvious  from  the  consideration  of  the  series  given  above  as 
examples.  It  is  not  difficult  to  prolong  these  series  until  their  terms  are  less 
than  Y^^  say,  or  x^Vo»  ^^d  from  this  it  seems  plausible  to  think  that  the 


SERIES  483 

terms  would  become  less  than  one  millionth,  or  one  billionth,  or  any  other 
number,  if  a  sufficient  number  of  terms  are  taken.  As  a  matter  of  fact 
this  is  true,  but  the  proof  is  too  difficult  to  be  given  here. 

III.   We  have  proved  that,  if  s„  denote  the  sum  of  the  first 
n  terms  of  a  geometric  series, 

(L  —  a?*" 

CI  ^      V       \ 

This  may  be  written :  s_  = ar"~V ). 

1  —  r  \l  —  rj 

By  taking  n  sufficiently  large,  the  product  of  ar''~'^  and  the 

fixed  number can  be  made  as  small  as  desired.     As  more 

1  —  r 

and  more  terms  of  the  series  are  added,  the  sum  differs  less 

and  less  from  ;  and  if  sufficient  terms  are  taken,  the  sum 

1  —  r 

comes  and  remains  as  close  as  we  please  to 


1-r 

The  number      ^      is   called   the   limit  of   the    sum   of  n 
1  —  r 

terms,  as  n  is  increased  without  bound.     Denoting  this  limit 
by  s,  we  have  : 

a 
s  = • 


The  number  s  is  not  the  sum  of  all  the  terms  of  the  series,  for  since  the 
terms  of  the  series  never  come  to  an  end,  the  operation  of  adding  them 
cannot  be  completed. 

According  to  Sec.  644, 

s  =  — ~  when  w  =  oo. 
1  —  r 

For  example : 

1  4 
When  a  =  4,  and  r  =  -  ,  then  the  limit  of  s  = =  8. 

2  1  —  i 
To  test  this,  we  form  successive  values  of  s^. 

si  =  4. 

52  =  6. 

53  =  7. 

54  =  7^. 

SB  =  7|. 
se  =  7|. 
It  appears  that  the  values  of  s„  approximate  more  and  more  closely  to 
8  as  w  is  increased. 


484  A  HIGH   SCHOOL   ALGEBRA 

WRITTEN     EXERCISES 

Find  the  limit  of  the  sum  of  the  series : 

1.    H-i  +  i+--.  3.    5  +  1+1  +  ^.... 

5.  Test  the  results  of  the  preceding  exercises  by  finding 
successive  values  of  s„. 

6.  In  an  infinite  geometric  series  s  =  2  and  r  =  |- ;  find  a. 

7.  Find  the  fraction  which  is  the  limit  of  .333333  •••,  or 
.3 +  .03 +  .003+.... 

8.  Find  the  limitof  .23232323  •••  or  .23 +  .0023 +.000023  +  ..-. 


9.  Triangles  are  drawn  in  a 
rectangle  of  dimensions   indi- 
A  B  D     E    c   cated,  B  being  the  midpoint  of 

DC,  and  so  on.     What  limit  does  the  sum  of  the  areas  of  the 
triangles  approach  as  more  and  more  triangles  are  taken? 

10.  Find  the  sum  of  16  terms  of  the  series, 

27,  221  18,  131,  .... 

11.  Find  the  sum  of  18  terms  of  the  series, 

36,12,4,1,.... 

12.  The  difference  between  two  numbers  is  48.  Their  arith- 
metical mean  exceeds  their  geometric  mean  by  18.  Find  the 
numbers., 

13.  Express  as  a  geometric  series  the  decimal  fraction. 

.0373737  .... 

What  is  its  limiting  value? 

14.  Find  the  limiting  value  of  each  of  these  series : 
(a)   .353535....  (e)   3.605605.... 
(6)   .125666....                        (/)  '5.00888  .... 

(c)  .032424....  {g)   9.161010.... 

(d)  .125125-...  Qi)  6.043838.... 


SERIES  485 

16.   If    ,  — -, ,   are   in   arithmetical   progression, 

5  — a    2h    h  —  c 

show  that  a,  b,  c  are  in  geometric  progression. 
Suggestion.    The  supposition  means  that 

_J L  =  J: L_. 

b-a     2b     26      b  -  c' 
This  reduces  to  b^  =  ac. 

16.  Find  the  amount  in  n  years  of  P  dollars  at  r  per  cent 
per  annum,  interest  being  compounded  annually. 

17.  During  a  truce,  a  certain  army  A  loses  by  sickness  14 
men  the  first  day,  15  the  second,  16  the  third,  and  so  on; 
while  the  opposing  army  B  loses  12  men  every  day.  At  the 
end  of  fifty  days  the  armies  are  found  to  be  of  equal  size. 
Find  the  difference  between  the  two  armies  at  the  beginning  of 
the  truce. 

18.  A  strip  of  carpet  one  half  inch  thick  and  29f  feet  long 
is  rolled  on  a  roller  four  inches  in  diameter.  Find  how  many 
turns  there  will  be,  remembering  that  each  turn  increases  the 
diameter  by  one  inch,  and  taking  as  the  length  of  a  circumfer- 
ence^ times  the  diameter. 

19.  Insert  between  1  and  21  the  arithmetic  means  such  that 
the  sum  of  the  last  three  terms  of  the  series  is  48. 

20.  If  -  =  -,  prove  that  ab-\~cd  is  a  mean  proportional  be- 
tween a^-\-c^  and  b^  +  dl 

21.  The  sum  of  the  first  ten  terms  of  a  geometric  series  is 
244  times  the  sum  of  the  first  five  terms ;  and  the  sum  of  the 
fourth  and  sixth  terms  is  135.  Find  the  first  term  and  the 
common  ratio. 

REVIEW 
WRITTEN     EXERCISES 

1.  Find  the  47th  multiple  of  7. 

2.  Find  the  sum  of  the  first  12  multiples  of  4. 


13. 

2,  4,  6, 

14. 

-5,- 

15. 

16      5 
^}  T'  T? 

1. 

l^>  TS' 

•  ••. 

4, 

-3, - 

10, 

i. 

-|>- 

-¥, 

i86  A    HIGH   SCHOOL   ALGEBRA 

Find  the  20th  term,  and  the  sum  of  12  terms  of  each  series : 

3.  6,  9,  ¥.-¥-.-• 

4.  8,11,  14,17,  .... 

5.  2^2«,  2^  .... 

6.  a  +  b,  a  —  b,  a  —  S  b,  a  —  5  b. 

Find  the  eighth  term,  and  the  sum  of  8  terms : 

7.  1,4,16,....  10.    1,  -2,22,  _23,  .... 

8.  3,6,12,....  11.    i,i  tV.  •••• 

9.  2,  -4,  8,  -16,  ....  12.   100,  -40,  16,  .... 

Find  the  twelfth  term,  and  the  sum  of  12  terms  : 

16. 

3,-1,....  17. 

18. 

19.  Find  three  numbers  whose  common  difference  is  1  and 
such  that  the  product  of  the  second  and  third  exceeds  that  of 
the  first  and  second  by  ^. 

20.  The  first  term  of  an  arithmetical  series  is  ri^  —  n  —  1, 
the  common  difference  is  2.     Find  the  sum  of  n  terms. 

21.  In  some  countries  of  Europe  the  hours  of  the  day  are 
numbered  on  the  clockface  from  1  to  24.  How  many  strokes 
would  a  clock  make  per  day  in  striking  these  hours  ? 

22.  How  many  strokes  does  a  common  clock  striking  the 
hours  make  in  a  day  ? 

23.  A  man  leases  a  business  block  for  20  years  under  the 
condition  that,  owing  to  estimated  increase  in  the  value  of  the 
property,  the  rental  is  to  be  increased  $50  each  year.  He 
pays  altogether  $19,500.  What  was  the  rental  of  the  first 
year  ?     The  last  ? 

24.  A  railroad  car  starting  from  rest  began  to  run  down  an 
inclined  plane.  It  is  known  that  in  such  motion  the  distances 
passed  over  in  successive  seconds  are  in  arithmetical  progres- 
sion. It  was  observed  that  at  the  end  of  10  sec.  the  car  had 
passed  over  570  ft.  and  at  the  end  of  20  sec.  2340  ft.  from  the 


SERIES  487 

starting  point.     How  far  did  it  run  the  first  second  ?     How 
far  from  the  starting  point  was  it  at  the  end  of  15  sec.  ? 

25.  It  is  known  that  if  a  body  falls  freely,  the  spaces  passed 
over  in  successive  seconds  are  in  arithmetical  progression,  and 
that  it  falls  approximately  16  ft.  in  the  first  second  and  48  ft. 
in  the  next  second.  To  determine  the  height  of  a  tower,  a  ball 
was  dropped  from  the  top  and  observed  to  strike  the  ground 
in  4  sec.     Find  the  height  of  the  tower. 

26.  An  employee  receives  a  certain  annual  salary,  and  in 
each  succeeding  year  he  receives  $72  more  than  the  year 
before.  At  the  end  of  the  tenth  year  he  had  received  all  to- 
gether S  10,440.  What  was  his  salary  the  first  year?  The 
last? 

27.  The  14th  term  of  an  arithmetical  series  is  72,  the  fifth 
term  is  27.     Find  the  common  difference  and  the  first  term. 

28.  A  man  is  credited  $100  annually  on  the  books  of  a 
building  society  as  follows :  At  the  beginning  of  the  first  year 
he  pays  in  $100  cash.  At  the  beginning  of  the  second  year 
he  is  credited  with  $6  interest  on  the  amount  already  to  his 
credit;  and  he  is  required  to  pay  $94  in  cash,  making  his 
total  credit  $200.  At  the  beginning  of  the  third  year  he  is 
credited  with  $  12  interest,  and  pays  $  88  in  cash,  and  so  on. 
How  much  is  his  payment  at  the  beginning  of  the  tenth 
year?  What  is  his  credit  then?  How  much  cash  has  he 
paid  in  all  ? 

29.  At  each  stroke  an  air  pump  exhausts  f  of  the  air  in  the 
receiver.  What  part  of  the  original  air  remains  in  the  receiver 
after  the  8th  stroke  ? 

30.  At  the  close  of  each  business  year,  a  certain  manufac- 
turer deducts  10  %  from  the  amount  at  which  his  machinery 
was  valued  at  the  beginning  of  the  year.  If  his  machinery 
cost  $  10,000,  at  what  did  he  value  it  at  the  end  of  the  fourth 
year? 

31.  In  Exercise  30,  by  use  of  logarithms,  find  its  valuation 
at  the  end  of  the  20th  year. 

32 


488  A   HIGH   SCHOOL   ALGEBRA 

32.  Show  that  the  terms  of  a  geometric  progression  form  a 
continued  proportion,  by  applying  Sec.  595  to  the  series  a,  ar, 
ar^,  ar^,  ar^,  •  •  • . 

33.  We  have  shown  in  Sec.  644  that  — —  is  the  limit  of 

1  — r 

the  sum  of  the  terms  of  a  geometric  progression  whose  first 
term  is  a  and  whose  common  ratio  is  7\  Find  the  terms  of 
this  series  by  dividing  the  numerator  of  the  above  fraction  by 
its  denominator. 

34.  What  kind  of  a  series  do  the  reciprocals  of  the  numbers 
2.  3,  and  6  form  ? 

35.  Find  the  arithmetical  mean  between  —  and  -. 

a  b 

36.  Find  the  sum  of  each  of  the  following  infinite  series : 

(1)  3  +  i  +  TV  +  --- 

(2)  l+i  +  i+J^  +  .... 

(3)  l  +  i  +  ^  +  .... 

37.  Find  the  sum  of  10  terms  of  each  series  in  Exercise  36. 

38.  Write  the  7'th  term  of  the  binomial  expansion ;  also  the 
(r  +  l)st  term.  Find  the  ratio  of  the  rth  term  to  the  (r  -f-  l)st 
term,  and  simplify  the  result. 

39.  Write  the  term  which  contains  x^^  in  the  expansion  of 
(a  -  xy\ 

SUMMARY 

The  following  questions  summarize  the  definitions  and  proc- 
esses treated  in  this  chapter : 

1.  When  is  a  group  of  numbers  called  a  se7^iesf         Sec.  612. 

2.  What  is  meant  by  the  terms  of  a  series  ?  Sec.  613. 

3.  Define  and  illustrate  an  arithmetical  series,  or  progression. 

Sec.  616. 

4.  Define  common  difference.  Sec.  617. 

5.  Define  and  illustrate  a  geometric  progression.         Sec.  623. 

6.  Define  common  ratio.  Sec.  624. 

7.  De^ne  arithmetical  mean  ;  geometric  mea,n.    Sees.  631-633- 


SERIES  489 

8.  State  the  formula  for  the  last  term  or  general  term  of  an 
arithmetical  progression ;  also  of  a  geometric  progression. 

Sees.  619,  626. 

9.  State  the  formula  for  the  sum  of  n  terms  of  an  arithmet- 
ical progression ;  also  of  a  geometric  progression. 

Sees.  620,  627. 

10.  State  a  formula  to  find  n  in  terms  of  a,  I,  and  s  in  an 
arithmetical  progression.  Sec.  637. 

11.  State  a  formula  to  find  d  in  terras  of  a,  n,  and  I  in  an 
arithmetical  progression.  Sec.  637. 

12.  State  a  formula  to  find  a  in  terms  of  n,  I,  and  s  in  an 
arithmetic  progression.  Sec.  637. 

13.  State  a  formula  to  determine  a  in  terms  of  r,  n,  and  I  in 
a  geometric  progression.  '  Sec.  639. 

14.  State  a  formula  to  determine  r  in  terms  of  a,  Z,  and  s  in 
a  geometrical  progression.  Sec.  639. 

15.  State  the  general  form  of  the  rth  term  of  the  binomial 
expansion  (a±hY.  Sec.  641. 

16.  Define  and  illustrate  an  infinite  series.  Sec.  643. 

17.  State  the  expression  for  the  limit  of  s  in  an  infinite 
geometric  series.  Sec.  644,  III. 

HISTORICAL  NOTE 

The  existence  of  sets  of  successive  numbers,  each  term  of  which 
depends,  in  a  definite  way,  upon  its  predecessors  for  its  value,  called  a 
progression,  or  series  of  numbers,  was  discovered  by  the  early  mathema- 
ticians. One  of  the  two  Babylonian  tablets  still  in  existence  gives  in 
cuneiform  symbols  the  squares  of  the  integers  from  1  to  60,  namely,  1,  4, 

9,  16,  25,  and  so  on.     The  other  tablet  gives  the  following  numbers:  5, 

10,  20,  40,  80,  96,  112,  and  so  on,  the  first  five  of  which  form  a  geometric 
progression  and  the  rest  an  arithmetical  series.  These  numbers  were 
used  to  represent  the  illuminated  portions  of  the  moon's  disk  from  day  to 
day,  from  new  moon  to  full  moon.  Thus,  taking  the  whole  as  240  parts, 
the  visible  portion  of  the  moon  for  the  first  day  would  be  ^|^,  or  ^^  of 
the  whole  disk. 

The  terms  of  a  geometric  series  with  an  integral  ratio  increase  at  a 
very  rapid  rate,  and  the  earlier  mathematicians  seem  to  have  taken  much 


V4.' 


490  A   HIGH   SCHOOL   ALGEBRA 

interest  in  the  framing  of  problems  intended  to  exemplify  this.  Sixteenth- 
century  writers  of  arithmetic  collected  many  of  these  problems,  of  which 
the  following  are  typical : 

A  peasant  agreed  with  a  blacksmith  to  pay  him  for  shoeing  his  horse 
at  the  rate  of  one  cent  for  the  first  nail,  two  cents  for  the  second,  four 
cents  for  the  third,  and  so  on,  in  geometric  progression.  There  being  8 
nails  in  each  of  the  four  shoes,  how  much  was  the  peasant  to  pay  the 
blacksmith  ? 

Required  the  number  of  kernels  of  wheat  needed  in  order  to  place  one 
kernel  on  the  first  square  of  the  chessboard,  two  on  the  second,  four  on 
the  third,  and  so  on,  for  the  sixty-four  squares.  This  later  problem  was 
given  by  Masudi,  in  Meadows  of  Gold  (950  a.d.). 

The  process  of  summing  arithmetical  and  geometric  series  (Sees.  621, 
628),  and  the  methods  for  finding  any  required  term,  were  known  to  the 
Hindoos  and  appear  in  the  works  of  Bhaskara.  But  these  were  trifling 
beginnings  compared  with  the  part  played  by  the  almost  unlimited  variety 
of  series  used  in  modern  mathematics.  The  proof  of  the  binomial  expan- 
sion by  Newton,  the  ability  to  express  logarithms  by  series,  accomplished 
by  Mercator  (1668),  and  the  study  of  other  systems  of  infinite  series  has 
opened  vast  new  fields  of  mathematics. 


CHAPTER   XXXIII 
GEOMETRIC  PROBLEMS  FOR   ALGEBRAIC  SOLUTION 

The  problems  in  the  following  list  may  be  used  as  supple- 
mentary work  for  pupils  that  have  studied  plane  geometry. 
In  the  body  of  the  Algebra  numerous  problems  have  been 
given  applying  geometric  facts  which  the  pupil  has  learned  in 
the  study  of  mensuration  in  arithmetic.  In  the  following  list 
the  problems  contain  the  application  of  other  relations  and 
theorems  of  geometry,  and  typical  solutions  have  been  inserted 
to  suggest  to  the  pupil  the  method  of  attack. 

LINEAR   EQUATIONS.     ONE   UNKNOWN 

1.  In  a  given  triangle  one  angle  is  twice  another,  and  the 
third  angle  is  24°.     Find  the  unknown  angles. 

Solution.    Let  x  be  one  of  the  unknown  angles, 

then  2  X  is  the  other.  (i) 

Because  the  sum  of  the  angles  of  a  triangle  =  180°, 

X  +  2  a:  4-  24°  =  180°.  {2) 

Solving  equation  (2) 

X  =  62°  and  2  X  =  104°.  (5) 

The  angles  of  the  triangle  are  52°,  104°,  24°. 

2.  In  a  certain  triangle  one  angle  is  three  times  another, 
and  the  third  angle  is  36°.     Find  the  unknown  angles. 

3.  In  a  given  right-angled  triangle  one  acute  angle  is  |  the 
other.     Find  the  angles. 

4.  In  a  certain  isosceles  triangle  the  angle  opposite  to  the 
base  is  18°.     Find  the  angles  at  the  base. 

5.  The  three  angles  A,  B,  and  C  of  a  given  triangle  are  in 
the  ratio  of  2,  3,  and  5.     Find  the  angles. 

491 


492 


A  HIGH  SCHOOL  ALGEBRA 


6-    Given  an  angle  A  such  that  a  point  B  situated  on  one 
side  11  in.  from  the  vertex  is  8  in.  distant  from  the  other  side. 
Find  a  point  C  on  the  same  side  as  B,  and 
equidistant  from  B  and  the  other  side  of  the 
angle. 

Solution.     Let  A  be  the  given  angle,  then  the 
figure  represents  the  conditions  of  the  problem. 
From  the  similar  triandes  A  CE&nd  ABD,  we  have 


{2) 
(5) 


From  (1), 

Then, 

The  distance  CB  is  ff  in.,  or  4^f. 


AC  _AB 
CE     BD' 

11  -  X       11 

X           8 

88-8a;  =  llx 

88  = 

=  19  X  and  X  =  f|. 

7.  Solve  problem  6,  if  the  point  JB  is  9  in.  from  ^  and  6  in. 
from  side  AD. 

8.  Solve  problem  6,  if  the  point  B  '\?,a  in.  from  the  vertex 
of  the  angle  and  h  in.  from  the  other  side  of  the  angle. 

9.  Two  points  A  and  B  are  8  in.  apart.  Parallels  are 
drawn  through  A  and  B ;  on  these  parallels  the  points  A^  and 
B^  are  located  on  the  same  side  of  the  straight  line  through  AB 
and  at  distances  6  in.  and  5  in.  from  A  and  JB,  respectively. 
Determine  the  point  where  the  line  A'B^  cuts  the  line  AB. 


Solution.     Let  C  be  the  desired  point  and  let  ^C  =  x. 
Then,  by  similar  triangles, 


BC 
BB' 

AC 
~AA'' 

X 

_x+S 

5 

6 

(2) 


GEOMETRIC   PROBLEMS  493 

Hence,  6x  =  6x  +  ^0.  (5) 

and  X  =  40.  i4) 

The  point  is  40  in.  from  B. 

10.  Solve  the  same  problem  if  A'  and  B'  lie  on  opposite 
sides  of  AB. 

11.  Solve  the  same  problem  if  the  distance  AB  is  d,  and 
the  points  A',  B'  lie  on  the  same  side  of  AB,  and  at  distances 
a  and  b  from  A  and  B,  respectively,  with  a  >  b. 

12.  Solve  the  preceding  problem  if  the  points  A'  and  B'  lie 
on  opposite  sides  of  AB. 

13.  The  three  sides  of  a  triangle  are  11,  9,  12.  A  perpen- 
dicular is  dropped  on  the  side  of  length  11  from  the  opposite 
vertex.  Find  the  lengths  of  the  segments  into  which  the  foot 
of  the  perpendicular  divides  that  side. 


Solution.     Using  the  notations  of  the  figure, 

h^  =  122  -  x^  (i) 

and  Zi2^92-(ll_x)2.  (£) 

From  (1)  and  (S),  122  _x'^  =  9^-  (11  -  x)^.  (3) 

Rearranging  (5),  122  _  92  +  112  =  22  x.  (4) 

Solving  (^),  x  =  H.  (5) 

The  other  segment  is  11  —  a;,  or  f  f-.  (6) 
The  segments  are  ff  and  ff . 

14.  Solve  the  preceding  problem  if  the  sides  are  4,  7,  9,  and 
the  perpendicular  is  dropped  on  the  side  of  length  7. 

15.  Solve  the  same  problem  if  the  sides  of  the  triangle  are 
a,  b,  c,  and  the  perpendicular  is  dropped  on  the  side  of  length  a. 


494 


A  HIGH   SCHOOL   ALGEBRA 


16.  The  lower  base  of  a  trapezoid  is  12,  the  upper  base  is  10, 
and  the  altitude  is  4.  Determine  the  alti- 
tude of  the  triangle  formed  by  the  upper 
base  and  the  prolongation  of  the  two  non- 
parallel  sides  until  they  meet. 

Solution.     Using  the  notations  of  the  figure, 

EF_DC  .J. 

EG'AB'  ^^ 

X+4:         12  ^    ^ 

Hence,  12  a;  =  10  a;  +  40,  (5) 

or  x=  20.  (^) 


/ 

/ 

/ 

E 

\ 

\\ 
\ 

i    \ 

r' 

^lO 

1     •! 

: 

^V                                           ^r 

12 


17.  Solve  the  same  problem  if  the  lower  base  is  a,  the  upper 
base  &,  and  the  altitude  h. 

LINEAR  EQUATIONS.     TWO  UNKNOWNS 

18.  A  rectangle  5  in.  longer  than  it  is  wide  is  inscribed  in  a 
triangle  of  base  12  in.  and  altitude  9  in.,  the  longer  side  rest- 
ing on  the  base  of  the  triangle.  Find  the  dimensions  of  the 
rectangle. 

Solution.    Let  x  denote  the  longer  side  and  y  the  shorter 

Then,  x  -  y  =  5.  (i) 


E 

1 

A 
F             \ 

D 

1 

V 

L 

y 

X 

\ 

U{                                                           jtJ 

In  the  similar  triangles  ABC  and  AED, 

AF^ED 
AG     BC' 

0  -  y  _  a:  . 
9         12* 


(5) 


GEOMETRIC   PROBLEMS 


495 


From  (3), 

From  (i)  and  (^), 


10S-12y  =  9x. 
108-  12  2/ =9(2/ +  5), 
21  y  =  63. 

y  =  3. 

x  =  8. 


(6) 


r> 

/t 

c 

£7 

/ 

20^ 

F 
G 

1/        N. 

a(          -                                     Ji^ 

From  (7)  and  (1), 

The  dimensions  of  the  rectangle  are  3  in.  and  8  in. 

19.  Solve  the  preceding  problem  if  the  shorter  side  rests  on 
the  base  of  length  12  in. 

20.  Solve  Problem  18  if  the  difference  of  the  sides  is  d,  the 
length  of  the  base  of  the  triangle  is  a,  the  altitude  is  7i,  and 
the  longer  side  of  the  rectangle  rests  on  the  base  of  the  triangle. 

21.  Solve  the  preceding  problem  if  the  shorter  side  of  the 
rectangle  rests  on  the  given  base  of  the  triangle. 

22.  A  rectangle  similar  to  a  rec- 
tangle whose  sides  are  5  and  8  is 
inscribed  in  a  triangle  of  base  32 
and  altitude  20.  The  longer  side 
of  the  rectangle  rests  on  the  given 
base  of  the  triangle.  Find  the  di- 
mensions of  the  rectangle. 

Solution.     Let  x  and  y  denote  the  sides  of  the  inscribed  rectangle. 
Then  from  the  similar  triangles  EBC  and  ABC^ 

or  -^  =r ""  ~  y .  (^) 

(3) 

(4) 

(^) 

(^) 

or  64  V  =  640.  (7) 

(9) 

23.  Solve  the  preceding  problem  if  the  shorter  side  of  the 
rectangle  rests  on  the  given  side  of  the  triangle. 


DE      CF 

AB      CO' 

X       20  —  y 
32         20 

From  the  similarity 

of  the 

rectangles, 

From  (5), 
From  (^), 

X     8 

~y'  I' 

x  =  \y. 
20  a;  =  640  -  32  y. 

From  (4)  and  (5), 

S2y  =  640  -  32  y, 
64  y  =  640. 
2/ =  10. 

From  (S)  and  (4), 

x  =  l6. 

496  A  HIGH  SCHOOL  ALGEBRA 

24.  Solve  the  same  problem  if  the  given  side  of  the  triangle 
is  of  length  a,  and  the  altitude  on  it  is  of  length  h,  and  the 
given  rectangle  has  dimensions  I  and  m,  provided  the  inscribed 
rectangle  has  its  side  corresponding  to  the  side  I  of  the  given 
rectangle  resting  on  the  given  base  of  the  triangle. 

25.  The  bisector  of  an  angle  of  a  given  triangle  divides  the 
side  opposite  to  the  angle  into  two  segments  of  lengths  4  in. 
and  7  in.  The  difference  between  the  other  two  sides  of  the 
triangle  is  5  in.     Find  the  perimeter  of  the  triangle. 


Solution.     Let  ABC  be  the  given  triangle,  AD  the  bisector  of  angle 

A,  and  x  and  y  the  required  sides. 

Then,  x  —  y  =  6,  given  in  the  problem,  (i) 

r      7 
and-  =  -,  by  geometry  the  bisector  divides  the  opposite  side  into  seg- 
y      4 

ments  proportional  to  the  adjacent  sides.  (^) 

4  5C  =  7  y,  from  (^) .  *  (5) 

7  y-4y  =  20,  from  (3)  and  4  times  (i) .  (4) 

Then,  y  =  6f,  solving  (^).  (5) 

Then,  x  =  U^,  from  (5)  and  (i).  (6) 

The  perimeter  is  11  in.  +  6f  in.  +  llf  in.  =  29J  in. 

26.  Solve  the  preceding  problem,  if  the  segments  of  the  base 
are  I  and  m  and  the  difference  between  the  sides  is  d.  Also,  if 
the  segments  of  the  base  are  p  and  q  and  the  difference  between 
the  sides  is  r. 

27.  The  sides  of  a  triangle  are  8  ft.,  12  ft.,  and  15  ft.,  and 
the  angle  between  the  sides  8  and  12  is  bisected  by  a  line  cut- 
ting the  side  15.  What  is  the  length  of  each  segment  of  the 
line  15  ? 


GEOMETRIC   PROBLEMS 


497 


LINEAR  EQUATIONS.     THREE  UNKNOWNS 

28.    The  points  A,  B,  and  C  are  situated  so  that  AB  =  8  in. 
BC=6  in.,  AC  =5  in.     Find  the 
radii   of  three   circles   having  the 
three   points   as  centers    and  each 
tangent  to  the  other  two  externally. 

Solution.    Let  x,  y,  0,  be  the  radii  of 
the  three  circles  as  indicated  in  the  figure. 

Then,  x  +  y  =  S,  {1)  B* 

x-\-  z=5,  {£) 

y  +  z  =  Q.  (5) 


Adding  (i),  (^),  and  (5),  and  dividing  the  result  by  2, 

x-{-y  +  z  =  Y-- 
From  {3)  and  (4),  x  -  |. 

From  {2)  and  (^),  y  =  |. 

From  (i)  and  (^),  z  =  |. 

The  radii  are  3|  in.,  4^  in.,  and  1^  in. 


{4) 

{5) 
{6) 
(7) 


29.    Solve    the    same   problem   if   AB 
AC  =  20. 


12,   50=16,    and 


30.    Solve  Problem  28,  if  BC=  a,  AC  =  b,  AB  =  c. 

31.  A  triangle  ABC  is  circumscribed 
about  a  circle  and  its  sides  are  tangent  at 
points  E,  D,  F.  The  sides  of  the  triangle 
are  a  =  8  in.,  b  =  lo  in.,  and  c  =  12  in.  Find 
the  segments  into  which  points  E,  D,  F 
divides  the  sides. 

Solution.  The  tangents  from  an  external  point 
are  equal,  hence  in  the  figure, 

a: +  2/ =8,  •  (i) 

x  +  «  =  15,  {2) 

y  +  z=12.  (3) 

Subtracting  (3)  from  (2),      x-y  =  S.  (4) 

Adding  (1)  and  (4)  2x  =  11  (5) 

.-.     X  =  6^. 
From  (1)  and  (2)       i/ =  2^  and  2;  =  9^. 


498 


A  HIGH   SCHOOL  ALGEBRA 


32.  Solve   the  same  problem,  if   a  =  24  ft.,  6  =  10  ft.,  and 
0=24  ft. 

Also,  if  a  =  30  yd.,  6  =  40  yd.,  c  =  50  yd. 

33.  A  man  owned  a  triangular,  unfenced  field  of  sides 
600  yd.,  700  yd.,  and  800  yd.  He 
sold  a  triangular  piece  cut  off  by  a 
straight  line  parallel  to  the  side  of 
length  800  yd.  and  found  that  1800 
running  yards  of  fence  would  be  re- 
quired   to    inclose    what    remained. 

Yb  Find  the  lengths  of  the  sides  of  the 
portion  sold. 
Solution.     Using  the  notations  of  the  figure, 

AB  -{-  BC  +  AC  -{-  2  DE  =  AB  +  BE  +  EG  +  CD  +  DA  -\-  2  DE 
=  (AB  +  BE+ED  +  DA)-\-iEC  +  CD -\-  DE). 
2100  +  2  X  =  1800  +{x-\-y+z). 
CD^DE 
CA      AB' 
CE^DE 
CB     AB' 


Hence, 
But 


and 


and 


Hence, 

From  (7)  and  (5), 
From  {9)  and  (^), 
Hence, 

From  {12)  and  (7), 
From  {12)  and  {8), 


y 

700 

z 

600" 


x-{-  y  +  z 
2100  +  2x 


_x_ 
800' 

X 

800 ' 

Ix 

■  8  ' 

.3^ 

4  ■ 

2\x 

8 
;  1800  + 


21a; 


The  lengths  of   the   sides  are 
CD  =  420  yd. 


—  =  300. 

8 


X  =  480. 
y  =  420. 
z  =  360. 

DE  =  480  yd.,  EC 


{2) 
{3) 

(4) 
{5) 
{6) 

(5) 

(5) 

{10) 


{13) 

(U) 

360   yd.,  and 


GEOMETRIC   PROBLEMS 


499 


34.  Solve  Problem  33  if  the  division  line  runs  parallel  to 
the  side  of  length  TOO  yd. 

35.  Solve  Problem  33  if  the  division  line  runs  parallel  to 
the  side  of  length  600  yd. 

36.  Solve  Problem  33  if  the  sides  are  a,  b,  c,  with  the  divi- 
sion line  parallel  to  a,  and  the  required  length  of  fence,  2 p. 

37.  It  is  known  that  the  sum  of  the  three  angles  of  any 
triangle  is  180°.  In  a  certain  triangle  the  difference  between 
the  first  angle  and  the  second  is  10°,  and  between  the  second 
angle  and  the  third  is  25°.  Find  the  number  of  degrees  in  each 
angle. 

38.  The  sum  of  the  dimensions  of  a  rectangular  box  is 
13^  ft. ;  the  height  equals  one  half  the  sum  of  the  length  and 
breadth  and  also  equals  twice  their  difference.  Find  the 
dimensions. 


QUADRATIC  EQUATIONS 

39.    The  sides  of  a  triangle  are  AC=  7,  BC 
Calculate  the  length  of  the  altitude 
on  the  side  10,  and  of  the  two  seg- 
ments into  which  the  altitude  divides 
that  side. 

Solution.    Using   the  notations  of   the 
figure, 


and 

Then, 
or 

Then, 
and 
also, 

By  (1), 

Then, 


9,  and  ^B  =  10. 


h^  = 

:72- 

x\ 

h^  = 

:92- 

(10- 

xy. 

72  _ 

-X2  = 

r92- 

(10- 

x)% 

72  = 

:92- 

100  +  20  X. 

68  = 

:20  a; 

» 

X  = 

■-V-; 

10 

—  X  = 

=  ¥• 

h  - 

72  _ 

(V)^ 

■  52- 

172 

.6\/26. 

{1) 

{2) 
{3) 
{4) 
{5) 
(6) 

(9) 


The  segments  are  -1  and  —  and  the  altitude  is  ' 

6  6  5 


500 


A   HIGH   SCHOOL  ALGEBRA 


40.  Calculate  similarly  the  length  of  the  altitude  on  the 
side  of  length  7,  and  of  the  segments  into  which  the  altitude 
divides  that  side. 

41.  Calculate  similarly  the  length  of  the  altitude  on  the 
side  of  length  9,  and  of  the  segments  into  which  the  altitude 
divides  that  side. 

42.  If  the  lengths  of  the  sides  of  a  triangle  are  a,  h,  c,  calcu- 
late the  lengths  of  the  segments  into  which  each  side  is  divided 
by  the  altitude  on  that  side. 

43.  In  a  circle  of  radius  10,  a 
chord  is  drawn  at  distance  6  from 
the  center.  Find  the  radius  of  a 
circle  that  is  tangent  to  the  circle, 
to  the   chord,  and  to  a  diameter 


10 

perpenaicuiar  ro  it;. 

Solution. 

Using  the  notations  of  the  figure : 

EG''  =  EF^  +  FC'\ 

(^) 

(10-a;)2  =  a:2a.(6  +  x)2. 

(2) 

100  -  20  X  +  x2  =  x2  +  36  +  12  X  +  x^. 

(3) 

x2  +  S2x-6^  =  0. 

(4) 

32±V322  +  4.64 
''-                     2 

{5) 

32  ±  16  V4  +  1 

^                                            2 

(^) 

^  _  16  ±  8  V5. 

{7) 

The  negative  value  of  x  being  inadmissible  under  the  geometric  condi- 
tions, we  have : 

44.  Solve  the  same  problem,  if  the  radius  of  the  given  circle 
is  12  ft.  and  the  chord  is  4  ft.  from  the  center. 

45.  Solve  Problem  43,  if  the  radius  of  the  given  circle  is 
r  and  the  chord  is  at  d  distance  from  the  center. 


GEOMETRIC   PROBLEMS 


501 


46.  A  point  P  is  selected  on  a  diameter  of  a  circle  of  radius 
6,  at  the  distance  1  from  the 
center.  At  P  a  perpendicular 
is  erected  to  the  diameter  in 
question,  and  a  tangent  is 
drawn  to  the  circle  such  that 
the  point  of  contact  of  the 
tangent  bisects  the  segment  of 
the  tangent  lying  between  the 
perpendicular  and  the  diame- 
ter produced,  Eind  the  dis- 
tance from  the  center  to  the 
point  where  the  tangent  cuts 
the  diameter  produced. 

Solution.     Let  CE  =  x,  (i) 

,  Then  in  the  right  triangle  CDE, 

be"  =  x2  -  62.  {2) 

{3) 

(4) 
(5) 

(6) 
(n 

(.5) 

or  x'^  -X  =2  x'-^  —  72.  (9) 

(10) 

ill) 

{12) 


From  the  similar  triangle  DO^  and  DGE, 

DE  _ 

CE 

_GE 
~  DE' 

DE^^ 

-CE'GE 

= 

-.x-GE. 

Since 

D  bisects  FE, 

0E  = 

_PE 

2 

x-1 

2 

From 

(2),  (5),  and  (7), 

x{x-l)_ 

2 

a^2_62, 

x'^  —  X  = 

2  x2  -  72. 

.-.  a;2 

+  x-72  = 
X  = 

0. 

1  _L.  VI  +  288 

=  -  9,  8. 


The  negative  root  also  indicates  a  solution.  Tt  means  that  a  second 
tangent  satisfying  the  required  conditions  cuts  the  diameter  produced  on 
the  opposite  side  from  P,  at  the  distance  9. 


502  A   HIGH   SCHOOL   ALGEBRA 

47.  Solve  the  same  problem  if  the  radius  is  12  and  the 
point  lies  at  the  distance  2  from  the  center. 

48.  Solve  the  same  problem  if  the  radius  is  15,  and  the  dis- 
tance of  P  from  the  center  is  35. 

49.  Solve  the  same  problem  if  the  radius  is  r  and  the  dis- 
tance from  the  center  is  d. 

50.  The  tangent  to  a  circle  is  a  mean  proportional  between 
the  segments  of  the  secant  from  the  same  point.  Find  the 
length  of  the  tangent,  if  the  segments  of  the  secant  are  4  ft. 
and  9  ft. 

51.  Two  chords  AB  and  CD  intersect  at  0  within  the  circle. 
The  product  of  OA  and  OB  equals  the  product  of  OC  and  OD. 
Given  OA  =  4,  0B=  8,  and  CD  =  12,  find  OC  and  OD. 

52.  4  times  the  square  of  the  altitude  (h)  of  an  equilateral 
triangle  equals  3  times  the  square  of  a  side  s. 
Express  this  by  an  equation.  Solve  the  equa- 
tion for  s.     Also  for  h  in  terms  of  s. 

53.    Find  the  altitude  of  an  equilateral  tri- 
angle whose  side  is  20  in.,  using  1.7321  as  V3. 

54.  Find  the  side  of  an  equilateral  triangle,  if  7i  =  9V3  in. 

55.  The  square  on  the  hypotenuse  of  a  right-angled  triangle 
is  equal  to  the  sum  of  the  squares  on  the  other 
two  sides.     Express  this  relation  in  the  form 
of  an  equation,  using  the  letters  in  the  figure. 

56.  Find  the  length  of  the  hypotenuse  of  a  right  triangle 
the  other  two  sides  of  which  are  3  ft.  and  4  ft.  Also  of  one 
whose  other  two  sides  are  15  ft.  and  20  ft. 

57.  Solve  the  equation  0^  =  0^  -\-  b^  for  a ;  for  b. 

58.  Find  b  in  Exercise  57,  if  c  =  25  and  a  =  15. 
Similarly,  determine  the  numbers  to  fill  the  blanks.     To 

simplify  calculation  use  the  relation  c'^  —  a'^  =  (c  —  a)(c  +  a) : 
59.  60.  61.  62.  63.  64.  65. 

a.  15  7  40  45  208  44 

b.  —  24  —  —  171  117 

c.  17  _  41  53  221  233  — 


GEOMETRIC   PROBLEMS 


503 


SIMULTANEOUS   QUADRATICS 

66.  The  owner  of  a  triangular  lot  whose  sides  are  70,  88, 
and  140  rd.  in  length  wishes  to  divide  it  by  a  straight  fence 
into  two  parts  that  shall  be 
equal  in  area  and  also  have  the 
same  perimeter.  If  the  fence 
connects  the  sides  of  length  70 
and  140  rd.,  how  must  it  be 
placed  ? 

Solution.     Let  DE  be  the  desired 
position  of  the  fence. 

Tlien,  A  ABG  =  2  A  DBE. 

Since  the  triangles  have  one  angle  in  common, 
A  ABC      70  .  140 


Av 


(^) 


or,  by  (1;, 


A  DBE 

2: 


xy 
70  .  140 


xy 

or  xy  =  S5-  140. 

By  the  conditions  of  the  problem, 

BD  +  BE  +  DE  =  DA  +  AC+CE  +  ED. 


(4) 

(5) 

Subtracting  DE  from  both  members  and  replacing  the  other  lines  by 
their  values, 

a;  +  y  =  70  -  y  +  140  -  X  +  88,  (6) 

2(x  +  y)  =  298,  (7) 

x  +  y  =  149.  {8) 

Erom  (4),  4  a;?/  =  4  •  35  •  140  =  1402.  {9) 


Squaring  (8)  and  subtracting  (9)  from  the  result, 
(x  -  yY  =  149-^  -  140^ 

=  (149 +  140)  (149- 
=  289  •  9. 
Then,  x- y  =±11  -  S  =±  61. 

From  (8)  and  (13),  x  =  100,  49. 

y  =  49,  100. 


140) 


(10) 
(11) 
(12) 
(13) 
(U) 
(15) 


These  values  satisfy  the  algebraic  equations,  but  in  the  concrete  prob- 
lem y  =  100  is  inadmissible,  since  y  lies  on  the  side  of  length  70.  Hence, 
in  the  concrete  problem,  the  result  is  a;  =  100,  y  =  49. 

67.    Solve  the  same  problem  if  the  fence  connects  the  sides 
of  length  70  and  88. 
33 


504 


A   HIGH   SCHOOL   ALGEBRA 


Then, 


68.  Solve  the  same  problem  if  the  fence  connects  the  sides 
of  length  88  and  140. 

69.  Solve  the  same  problem  if  the  sides  are  of  length  a,  b, 
c,  and  the  fence  connects  the  sides  of  lengths  a 
and  c. 

70.    Find  the  sides  of  a  right-angled  triangle, 
given  its  area  25,  and  its  perimeter  30. 

Solution.     Let  x,  y,  z  denote  the  sides  of  the  triangle, 
z  being  the  hypotenuse. 

a:2  +  ?/2  =  02^ 
X  +  y  +  ;2  =  30, 

2 
Multiplying  both  members  of  (3)  by  4, 

2  a:?/  =  100. 
Adding  (4)  and  (1),     x2  +  2  a;?/  +  y^  =  ^2  ^  100, 

(X  +  2/)2  zz:  ^2  ^  100. 

X  +  ?/---  ao  —  z, 

(X  +  2/)2=:(.SO-0)2. 
(30  -  ^)2  =  ^2  _^  100. 

9OO-6O0  +  ;22  =  ^2_^  100. 
60  0  =  800. 

X2  +  2  Xy  +  y2  ^  ^S^OJL. 


and 


From  (2), 

From  (8)  and  (6), 


From  (7), 


or 


Multiplying  (4)  by  2  and  subtracting  the  result  from  (14), 


2xy +  2/2 


,  10  Vf 


Adding  (13)  and  (16)  and  dividing  the  result  by  2, 

25  ±  5  V7 

^  =  —3 

Subtracting  (16)  from  (13)  and  dividing  the  result  by  2, 

25T5V7 


The  sides  are 


25  +  5\/7      25 


y 

5V7 


3 


,  and 


3 

40 


(-4) 

(^) 

(i^) 

(i^) 
(i5) 


(i5) 
(i5) 

(i7) 
{18) 


71.    Solve  the  same  problem  if  the  area  of  the  triangle  is  64, 
and  the  perimeter  48. 


INDEX 

The  numbers  in  the  Index  refer  to  pages  in  the  Book 


Abscissas,  204. 

Absolute  Terms,  168,  328. 

Addition,  46,  291. 

of  Fractions,  148. 

of  Imaginaries,  398. 

of  Monomials,  46. 

of  Polynomials,  49. 

of  Radical  Expressions,  261. 

Associative  Law  of,  60,  293. 

Commutative  Law  of,  49,  292. 
Ahmes,  74. 

Algebraic  Expressions,  2. 
Algebraic  Sum,  46. 
Algebraic  Symbols,  2. 
Al-Khowarazmi,  93,  281. 
Alternation,  193,  447. 
Antecedent,  181. 
Approximations,  263,  270. 

Roots,  270. 

Surds,  263. 
Associative  Law,  60,  293,  301. 
Axes,  203. 

Base,  9,  384. 
Bhaskara,  45,  281. 
Binomial  Equations,  435. 
Binomial  Expansion,  239,  479. 

Coefficients  in,  240. 

Formula  for,  241,  479. 
Binomials,  12. 

Squares  of,  103,  113. 
Braces  and  Brackets,  12. 

Canceling,  85,  145.      • 
Cardan,  45,  451. 
Characteristic,  384,  385. 
Checking.     See  Testing. 
Circle,  14. 
Coefficients,  10. 

Binomial,  240. 

Decimal,  171. 

Detached,  304,  306. 

Numerical,  10. 
Common  Difference,  466. 
Common  Ratio,  469. 
Commutative  Law,   8,  49,   292, 
300. 


Complex  Numbers,  398. 
Composition,  193,  448. 

and  Division,  193,  448. 
Conjugate : 

Binomial,  264. 

Complex  Numbers,  399. 
Consequent,  181. 
Constants,  204. 
Coordinates,  204. 

Origin  of,  203. 
Cube  Root,  110. 

of  Arithmetical  Numbers,  371. 

of  Polynomials,  373. 
Cubes : 

of  Binomials,  107. 

Sum  or  Difference  of,  122. 

Degree,  17. 

of  Equations,  167,  328. 

of  Monomials,  17. 

of  Polynomials,  17. 
Descartes,  20,  45,  209. 
Detached  Coefficients,  304,  306. 
Difference,  54. 

of  Two  Squares,  115, 117. 
Diophantos,  7, 20, 74, 93, 134, 281. 
Discriminant,  429. 
Distributive  Law,  78,  301. 
Division,  84,  305. 

of  Fractions,  158. 

of  Imaginaries,  400. 

of  Monomials,  84,  86. 

of  Polynomials,  88. 

Law  of  Exponents,  85,  356. 

Law  of  Signs,  40. 

Elimination,  212. 
Equations,  21,  65,  94,  128,  167, 
.      327. 

of  One  Unknown,  169,  327. 

of  Two  Unknowns,  211,  336. 

of  Three  Unknowns,  228,  343. 

of    More    Than    Three    Un- 
knowns, 230,  343. 

Binomial,  435. 

Conditional,  22,  327. 

Contradictory,  226,  341. 


605 


506 


INDEX 


Equations  : 

Degree  of,  167,  328. 

Dependent,  226,  342. 

Equivalent,  128,  328. 

Exponential,  393. 

Fractional,  168,  220. 

General,  173. 

Higher,  167,  288,  328,  434. 

Homogeneous,  284. 

Identical,  22,  327. 

Incompatible,  226,  341. 

Independent,  213,  343. 

Linear,  167,  328,  491,  494,  497. 

Literal,  217,  423. 

Members  of,  21. 

Properties  of,  26,  167. 

Quadratic,  131,  269,  345,  499. 

Radical,  275,  432. 

Roots  of,  25,  355. 

Simultaneous,   213,   283,   336, 
436,  503. 

Solving,  25,  27. 

Symmetrical,  285. 

Systems  of,  213,  336. 

Systems,  Indeterminate,  225. 
Evolution,  110,  243. 
Exponents,  9,  252,  355. 

Fractional,  255,  358,  367. 

Laws  of,  76,  355. 

Negative,  365,  367. 

Zero,  365,  367. 
Extremes,  190. 

Factoring,  109,  316. 

Applied  to  Equations,  128-134. 

General  Methods,  318-321, 431. 

Types,  110-125,  316. 
Factors,  8. 

Common,  136. 

Highest  Common,  136. 

Literal,  8. 

Numerical,  8. 

Prime,  9. 

Repeated,  75. 
Factor  Theorem,  322. 
Forms,  173,  217,  269,  345,  421.' 

Linear,  173,  217. 

Quadratic,  269,  345,  421. 
Formulas,  272,  337,  422,  469,  471, 

474. 
Fourth  Proportional,  191,  447. 
Fractions,  141,  309. 

Clearing  of,  170,  223. 

Common  Denominator  of,  147. 


Fractions  : 

Complex,  161,  315. 

In  Equations,  168. 

Law  of  Exponents  in,  145. 

Law  of  Signs  in,  142. 

Lowest  Common  Denominator 
of,  147. 

Lowest  Terms  of,  141. 

Numerator  and   Denominator 
of,  141. 

Processes  with,  148,  153,  310, 
314-315. 

Reduction  of,  145. 

Signs  of,  141,  150. 

Simple,  161. 

Terms  of,  141. 
Functions,  205. 

of  Dependent  Variable,  205. 

of  Independent  Variable,  205. 

Notation  of,  205. 

Gauss,  405. 
Gerbert,  290. 
Graphs,  34. 

of  Fractions,  310. 

of  Higher  Equations,  416. 

of  Imaginaries,  403. 

of  Integers,  291-295. 

of  Linear  Equations,  198. 

of  Quadratic  Equations,  407. 

of  Radical  Equations,  417. 

of     Simultaneous      Quadratic 
Equations,  414. 

of  Special  Systems,  341. 

Hamilton,  326. 

Harriot,  134. 

Historical  Notes,  7,  20,  45,  74,  93, 
134,  209,  250,  281,  290,  326, 
378,  39fc  405,  445,  489. 

Homogeneous  Equations,  284. 

Homogeneous  Expressions,  79. 

Identities,  22,,  327. 
Imaginary  Numbers,  397. 

Powers  of,  400. 

Processes  with,  398. 

as  Roots  of  Equations,  401. 
Indeterminate  Expressions,  458, 

459. 
Index,  257. 
Infinity,  459. 

Interpretation  of  Results,  226,347. 
Inversion,  192,  447. 


INDEX 


507 


Involution,  239. 
Irrational  Numbers,  252. 

Limits,  458. 
Linear  Equations. 

Solving,  169,  217,  228,  328. 
Linear  Forms,  173,  217. 
Logarithms,  379. 

Mantissa,  384. 

Mean  Proportional,  191,  193. 

Means,  190,  473. 

Arithmetical,  473. 

Geometric,  473. 
Monomials,  12. 

Degree  of,  17. 

Uses  of,  14. 
Multiples,  135,  138. 

Common,  138. 

Lowest  Common,  138. 
Multiplication,  75,  300. 

of  Fractions,  153. 

of  Imaginaries,  399. 

of  Monomials,  75,  76. 

of  Polynomials,  78,  80. 

of  Radicals,  262. 

of  Relative  Numbers,  39,  302. 
.  Associative  Law  of,  301. 

Commutative  Law  of,  8,  301. 

Distributive  Law  of,  78,  301. 

Law  of  Exponents  in,  76,  355. 

Law  of  Signs  in,  39,  77. 

Napier,  395. 

Negative  Numbers,  32,  294. 
Newton,  93,  250,  490. 
Notation,  1,  32,  183,  338. 

of  Positive  and  Negative  Num- 
bers, 32. 

Determinant,  338. 

Literal,  1. 

Special,  183. 
Numbers,  31.  . 

Complex,  398. 

Imaginary,  397. 

Irrational,  252. 
-Negative,  32,  294. 

Positive,  32,  294. 

Rational,  252. 

Real,  397. 

Relative,  31,  294. 

Signed,  33. 

Order  of  Operations,  10 
Ordinates,  204. 


Parentheses,  57,  60,  80. 

Uses  of,  11,  94. 
Peacock,  326. 
Plotting,  203. 
Polynomials,  12. 

Degree  of,  17. 

Uses  of,  16. 
Positive  Numbers,  32,  294. 
Power,  9. 
Prime  Factors,  9. 
Problems. 

of  Levers,  188,  449. 

of  Motion,  99. 

of  Similar  Triangles,  188. 

Clock,  334. 

Courier,  334. 

Geometric,  491. 

Planet,  334. 
Product,  8. 

Types,  102. 
Progression,  466,  469. 

Arithmetical,  466. 

Geometric,  469. 
Proportion,  185,  447. 
Proportional,  191,  447. 

Fourth,  191,  447. 

Mean,  191,  447. 

Third,  191,  447. 
Pro  Rata,  186. 

Quadrant,  204. 

Quadratic   Equations,    131,  269, 
345,  421,  499. 
Affected,  269. 
Complete,  269. 
General  Form  of,  131,  269,  345,  >' 

421. 
General  Solution  of,  272,  345, 

422. 
Incomplete,  132,  269. 
Literal,  217. 
Pure,  269. 

Simultaneous,  283,  503. 
Quadratic  Forms,  269,  345,  421. 
Quotients,  Special,  307. 

Radical  Expressions,  253. 

Addition  of,  261. 

in  Equations,  275,  432. 

Multiplication  of,  262. 

Properties  of,  253. 

Square  Root  of,  370. 

Subtraction  of,  261. 
Radicals,  252. 
Radicand,  253. 


508 


INDEX 


Ratio,  181. 

Common,  469. 

Greater  Inequality,  184. 

Less  Inequality,  184. 
Rational  Numbers,  252. 
Rationalizing  Factors,  264. 

the  Denominator,  264. 
Real  Numbers,  397. 
Reciprocal,  158,  315. 
Recorde,  20. 

Relative  Numbers,  31,  294. 
Roots  of  Equations,  25,  355. 

Approximate,  270. 

Imaginary,  401. 

Relation    of,    to    Coefficients, 
427. 

Symmetric  Functions  of,  427. 

Series,  465. 

Arithmetical,  466. 

Finite,  481. 

Geometric,  469, 

Infinite,  481. 

Sum  of,  467,  470. 
Signed  Numbers,  33. 
Signs,  1. 

of  Character,  32. 

of  Deduction,  28. 

of  Equality,  1. 

of  Fractions,  141,  150. 

of  Inequality,  41. 

of  Operation,  1,  32. 

of  Square  Roots,  109. 
Squares : 

of  Binomials,  103,  113. 

of  Polynomials,  104,  116. 

Completing,  269. 

Difference  of,  104, 115, 117, 320. 
Square  Root,  109,  243. 

of  Arithmetical  Numbers,  244. 

of  Polynomials,  247. 
Stevin,  395. 
Stifel,  20,  281. 
Subscripts,  183. 
Substitution,  25. 

Method  of,  215,  336. 
Subtraction,  54,  294. 

of  Fractions,  148. 

of  Imaginaries,  398. 

of  Monomials,  54. 

of  Polynomials,  55. 

of  Radicals,  261. 

Method  of,  213,  336. 


Summaries,  6,  19,  30,  44,  53,  64, 
83,  93,  134,  140,  166,  180, 
197,  209,  237,  250,  268,  280, 
289,  325,  354,  377,  395,  405, 
420,  444,  464,  488. 
Surds,  253. 

Biquadratic,  253. 

Cubic,  253. 

Entire,  253. 

Mixed,  253. 

Order  of,  253. 

Properties  of,  253. 

Quadratic,  253. 
Systems  of  equations. 

See  Equations. 

.Tartaglia,  451. 
Terms,  12. 

Absolute,  168. 

Compound,  12. 

Like,  46. 

in  Series,  465. 

Transposing,  69. 
Testing,  27,  50,  56,  78,  79,  89, 

428. 
Third  Proportional,  191,  447. 
Transposing  Terms,  69. 
Trinomials,     12,    118,   120,  318, 

431. 
Type  Products,  102. 

Unknown,  25.' 

Value,  3. 

Absolute,  33,  295. 

Numerical,  33. 
Values,  Tabulation  of,  4. 
Variables,  204. 

Dependent,  205. 

Independent,  205. 
Variation,  189,  452. 

Direct,  452. 

Inverse,  453. 

Relation  of,  to  Proportion,  189, 
452. 
Verifying.    See  Testing. 
Vieta,  250,  378. 

Wallis,  378. 
Wessel,  406. 

Zero,  308. 


(1) 


CHEMISTRY  FROM  A  NEW  STANDPOINT 

An  Inductive  Chemistry 

By  Robert  H.  Bradbury,  A.M.,  Ph.D.,  Head 
of  the  Department  of  Science,  Southern  High  School, 
Philadelphia.     i2mo.  Cloth,  $1.25. 

This  is  a  complete,  modern  text,  practical  and  teachable, 
for  high-school  students. 

The  subject  is  developed  along  lines  adapted  to  the  pupil's 
equipment.  Both  method  and  order  of  presentation  are  par- 
ticularly sound.  Constant  appeal  is  made  to  the  experience 
of  the  student's  everyday  life,  and  new  material  is  introduced 
in  logical  order,  according  to  the  interest  and  natural  com- 
prehension of  the  student. 

The  text  is  eminently  modern  in  spirit,  as  to  subject  matter. 
Recent  developments  of  the  first  importance,  absent  in  other 
texts,  are  simply  but  adequately  treated  here.  Emphasis  is 
likewise  laid  on  the  method  of  reasoning  followed  in  attaining 
results  as  well  as  on  the  mere  facts  of  the  science. 

The  book  is  unusually  teachable,  in  many  ways.  It  intro- 
duces the  history  of  the  subject  to  the  best  advantage.  It 
contains  helpful  classifications,  in  the  form  of  tables,  and 
lists  of  definitions.  The  illustrations,  too,  are  unusually 
clear  and  relevant. 

The  Syllabus  and  Entrance  Examination  requirements 
have  been  thoroughly  covered.  The  fact  that  these  required 
topics  are  printed  in  bold-faced  type  in  the  index,  will  be  of 
great  assistance  to  teachers  and  students. 

D.    APPLETON     AND     COMPANY 

NEW  YORK  CHICAGO 

sole 


TWENTIETH  CENTURY  TEXT-BOOKS 

A  High  School  Course  in  Physics 

By  Frederick  R.  Gorton,  Ph.D.,  Associate 
Professor  of  Physics,  Michigan  State  Normal 
College.  Fully  and  Practically  Illustrated.  i2mo, 
Cloth,  $1.25  net. 

The  author  has  written  a  text-book  in  Physics  which  is 
above  all  things  practical,  from  the  point  of  view  of  both 
teacher  and  pupil.  All  scientific  facts  are  clearly  stated  and 
referred,  wherever  such  reference  will  be  valuable,  to  the 
student's  own  experience  and  to  the  ordinary  phenomena  of 
his  everyday  observation.  Without  sacrificing  the  essentially 
scientific  nature  of  the  subject  -the  author  recognizes  it  as  a 
science  which  has  a  very  definite  bearing  upon  everyday  life. 

The  author  believes  that  the  class-room  work  should  be 
accompanied  by  suitable  laboratory  experimentation  by  the 
pupils  supplemented  with  demonstrative  experiments  by  the 
instructor. 

Stress  is  laid  upon  the  beneficial  results  to  which  the  study 
of  Physics  has  led  as  its  development  has  progressed.  Espe- 
cial attention  has  been  given  to  the  interesting  historical 
development  of  the  subject.  Portraits  and  adequate  bio- 
graphical sketches  of  many  scientists  to  whom  the  discovery 
of  great  principles  is  due  have  been  inserted. 

The  problems  throughout  the  book  eliminate  the  usual 
exercises  in  pure  reduction  and  substitute  those  of  a  more 
concrete  and  practical  nature. 

The  apparatus  described  is  as  simple  as  experience  has 
shown  to  be  consistent  with  satisfactory  results. 

The  illustrations  are  abundant  and  each  is  given  a  descrip- 
tive legend. 

To  aid  the  pupil  in  reviewing  and  the  teacher  in  quizzing, 
there  are  summaries  at  the  ends  of  the  chapters. 

No  subject  has  been  left  out  that  is  called  for  in  the  report 
of  the  College  Entrance  Requirement  Board. 

D.     APPLETON     AND     COMPANY 

NEW  YORK  CHICAGO 


ANCIENT  HISTORY  FOR  THE  HIGH  SCHOOL 

The  Story  of  the  Ancient  Nations 

By  William  L.  Westermann,  Associate  Pro- 
fessor in  History,  University  of  Wisconsin.  Illus- 
trated.    i2mo,  Cloth,  $1.50. 

There  is  no  other  branch  of  history  taught  in  our  High 
Schools  in  which  so  much  new  material  has  come  to  light 
during  recent  years  as  in  ancient  history.  Much  of  the  best 
source  material  is  not  available,  in  translated  form,  to  the 
teacher.  This  text-book  has  been  written  with  the  desire  to 
put  into  the  hands  of  High  School  teachers  and  pupils,  in 
simple  and  concrete  form,  the  story  of  the  development  of 
ancient  civiHzation  as  it  appears  in  the  light  of  the  historical 
material  recently  discovered.  It  is  the  outcome  of  more  than  a 
decade  of  teaching,  both  in  High  School  and  University  classes. 

The  attempt  has  been  made  to  present  the  progress  of 
ancient  civilization  as  a  continuous  and  unified  process. 
There  has  been  included,  in  simple  terms,  as  much  of  the 
business  and  social  background  as  space  would  permit. 

The  language  of  the  book  is  clear  and  succinct.  The  order 
of  presentation  is  logical,  and  the  correlation  of  facts  exact. 
There  are  exceedingly  helpful  and  well-written  generaliza- 
tions, giving  the  significance  of  the  various  periods. 

There  are  plentiful  maps  throughout  the  book.  The  illus- 
trations, with  the  exception  of  a  few  carefully  selected  restora- 
tions, are  almost  entirely  drawn  from  ancient  sources.  They 
have  been  carefully  chosen  for  the  light  which  they  throw  on 
the  life  of  the  people. 

D.    APPLETON     AND     COMPANY 

NEW  YORK  CHICAGO 

504e 


TWENTIETH   CENTURY  TEXTS  IN  ENGLISH 

Edited,  with  Intrcxluctions  and  Notes.    lUostrated.     J2mo.    Cloth. 
Addison  and  Steele's  The  Sir  Roger  de  Coverley  Papers  from 

The   Spectator.     Franklin  T.  Baker,  A.M.,  and  Richard  Jones,  Ph.D. 
207  pages.    30  cents. 

Browning's  Select  Poems.    HughC.  Laughlin,  a.m.   137  pages.  30 cents. 
Burke's  Speech  on  Conciliation  with  America.    William  i.  Crane. 

185  pages.     30  cents. 

Carlyle's  Essay  on  Burns.    Including  a  Selection  of  Burns'  Poems.    Carrie 
E.  T.  Dkacass.     166  pages.     3®  cents. 

Coleridge's   Rime  of  the  Ancient    Mariner,  and  Other  Poems. 

Pelhaim  Edgar,  B.A.,  Ph.D.     144  pages.     25  cents. 

Dryden's  Palamon  and  Arcite.     George  M.  Marshall,  Ph.B.     112  pages. 

25  cents. 
Eliot's  Silas   Marner.     J.  Rose  Colby,  Ph.D.,  and  Richard  Jones,  Ph.D. 
315  pages.     30  cents. 

Goldsmith's  The  Traveller  and  The  Deserted  Village.    Horatio 

N.  Drury.     96  pages.     30  cents. 
Goldsmith's  Vicar  of  Wakefield.     Louise  Maitland.    310  pages.   40  cents. 

Huxley's  Autobiography  and  Selected  Essays.    Sarah  E.  Simons, 

A.M.     248  pages.     40  cents. 
Lamb's   Selected   Essays.     Howard  Bement,  A.M.    341  pages.     50  cents. 

Macaulay's   Essays   on   Addison  and   Johnson.    George  B.  Aiton, 

A.M.     193  pages.     30  cents. 

Macaulay's  Essays  on  Milton  and  Addison.     George  b.  Aiton,  A.M. 

188  pages.     25  cents. 

Milton's    Shorter    Poems    and    Sonnets.      Frederick  D.  Nichols. 

153  pages.     25  cents. 
Scott's  Ivanhoe.     Carrie  E.  T.  Dracass.    621  pages.    60  cents. 
Scott's  Quentin  Durward.     J.  Rose  Colby,  Ph.D.    539  pages.    60  cents. 

Scott's    The   Lady   of  the  Lake.      James  Chalmers,  Ph.D.,  LL.D.     213 

pages.     30  cents. 

Shakspere's  Julius  Caesar,    w.  h.  McDougal.    158  pages.   35  cents. 

Shakspere's   Macbeth.     Richard  Jones,  Ph.D.     195  pages.    30  cents. 

Shakspere's  The  Merchant  of  Venice.      Richard  Jones,  Ph.D.,  and 

Franklin  T.  Baker,  A.M.     124  pages.     30  cents. 
Tennyson's  Idylls  of  the  King.      Joseph  V.  Denney.    144  pages.    30  cents. 

Tennyson's    The     Princess.       Franklin    T.    Baker,    a.m.       148    pages 

25  cents. 

Washington's  Farewell  Address  and  Webster's  First  Bunker 

Hill    Oration.      James  Sullivan,  Ph.D.     91  pages.     25  cents. 

D.     APPLETON     AND     COMPANY 

NEW  YORK  CHICAGO 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 
BERKELEY 

Return  to  desk  from  which  borrowed. 
This  book  is  DUE  on  the  last  date  stamped  below. 


.       2lOcr49B^^ 

16Gu 

im^  ■ 

I         24Jan  5  ICC 

90ct'55HJi 

3EP2  51355LTJ 


29^0* 


^TWH 


REC'D  L.U 
NOV  2  7  1957 

LD  21-100m-9,'48(B399sl6)476 


6May'58G^ 
REC'D  LD 

MAYUldSS 

REC'D  LD 


'^■i-i^'i 


264193 


UNIVERSITY  OF  CAIvlFORNIA.  IvIBRaRY 


